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Fibonacci sequence

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</style><div role="note" class="hatnote navigation-not-searchable">For the chamber ensemble, see <a href="Fibonacci_Sequence_(ensemble)" title="Fibonacci Sequence (ensemble)">Fibonacci Sequence (ensemble)</a>.</div>

In mathematics, the Fibonacci sequence is a <a href="Integer_sequence" title="Integer sequence">sequence</a> in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins

<dl><dd>0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... (sequence <a href="https://oeis.org/A000045" class="extiw external" title="oeis:A000045">A000045</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

The Fibonacci numbers were first described in <a href="Indian_mathematics" title="Indian mathematics">Indian mathematics</a> as early as 200 BC in work by <a href="Pingala" title="Pingala">Pingala</a> on enumerating possible patterns of <a href="Sanskrit" title="Sanskrit">Sanskrit</a> poetry formed from syllables of two lengths.<a href="#cite_note-GlobalScience-3">[3]</a><a href="#cite_note-HistoriaMathematica-4">[4]</a><a href="#cite_note-Donald_Knuth_2006_50-5">[5]</a> They are named after the Italian mathematician Leonardo of Pisa, also known as <a href="Fibonacci" title="Fibonacci">Fibonacci</a>, who introduced the sequence to Western European mathematics in his 1202 book <a href="Liber_Abaci" title="Liber Abaci">Liber Abaci</a>.<a href="#cite_note-FOOTNOTESigler2002404–05-6">[6]</a>

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the <a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a>. Applications of Fibonacci numbers include computer algorithms such as the <a href="Fibonacci_search_technique" title="Fibonacci search technique">Fibonacci search technique</a> and the <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> <a href="Data_structure" title="Data structure">data structure</a>, and <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> called <a href="Fibonacci_cube" title="Fibonacci cube">Fibonacci cubes</a> used for interconnecting parallel and distributed systems. They also appear <a href="Patterns_in_nature#Spirals" title="Patterns in nature">in biological settings</a>, such as branching in trees, <a href="Phyllotaxis" title="Phyllotaxis">the arrangement of leaves on a stem</a>, the fruit sprouts of a <a href="Pineapple" title="Pineapple">pineapple</a>, the flowering of an <a href="Artichoke" title="Artichoke">artichoke</a>, and the arrangement of a <a href="Pine_cone" class="mw-redirect" title="Pine cone">pine cone</a>'s bracts, though they do not occur in all species.

Fibonacci numbers are also strongly related to the <a href="Golden_ratio" title="Golden ratio">golden ratio</a>: <a href="#Binet's_formula">Binet's formula</a> expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to <a href="Lucas_number" title="Lucas number">Lucas numbers</a>, which obey the same <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> and with the Fibonacci numbers form a complementary pair of <a href="Lucas_sequence" title="Lucas sequence">Lucas sequences</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The Fibonacci numbers may be defined by the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a><a href="#cite_note-FOOTNOTELucas18913-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{0}=0,\quad F_{1}=1,}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}}</annotation>

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for n > 1.

Under some older definitions, the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0}">

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<annotation encoding="application/x-tex">{\displaystyle F_{0}=0}</annotation>

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</math><img src="./58ebe8b2d5551fb272cd4258940fe1e492592d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=0}" loading="lazy"> is omitted, so that the sequence starts with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=F_{2}=1,}">

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<annotation encoding="application/x-tex">{\displaystyle F_{1}=F_{2}=1,}</annotation>

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</math><img src="./39ce4b302203aa4afd0eccf11b8ccbb207fadd06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.104ex; height:2.509ex;" alt="{\displaystyle F_{1}=F_{2}=1,}" loading="lazy"> and the recurrence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}}</annotation>

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</math><img src="./4fa6d281e7a54e08aeffeef7458ddc0884333686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.279ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}" loading="lazy"> is valid for n > 2.<a href="#cite_note-FOOTNOTEBeckGeoghegan2010-8">[8]</a><a href="#cite_note-FOOTNOTEBóna2011180-9">[9]</a>

The first 20 Fibonacci numbers Fn are:

<th>F0

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<th>F1

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<th>F2

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<th>F3

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<th>F4

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<th>F5

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<th>F6

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<th>F7

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<th>F8

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<th>F9

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<th>F10

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<th>F11

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<th>F12

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<th>F13

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<th>F14

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<th>F15

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<th>F16

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<th>F17

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<th>F18

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<th>F19

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<td>0

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<td>1

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<td>1

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<td>2

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<td>3

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<td>5

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<td>8

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<td>13

</td>

<td>21

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<td>34

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<td>55

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<td>89

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<td>144

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<td>233

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<td>377

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<td>610

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<td>987

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<td>1597

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<td>2584

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<td>4181

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div class="mw-heading mw-heading3"><h3 id="India">India</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Golden_ratio#History" title="Golden ratio">Golden ratio § History</a></div>

The Fibonacci sequence appears in <a href="Indian_mathematics" title="Indian mathematics">Indian mathematics</a>, in connection with <a href="Sanskrit_prosody" title="Sanskrit prosody">Sanskrit prosody</a>.<a href="#cite_note-HistoriaMathematica-4">[4]</a><a href="#cite_note-knuth-v1-10">[10]</a><a href="#cite_note-FOOTNOTELivio2003197-11">[11]</a> In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm+1.<a href="#cite_note-Donald_Knuth_2006_50-5">[5]</a>

Knowledge of the Fibonacci sequence was expressed as early as <a href="Pingala" title="Pingala">Pingala</a> (<abbr title="circa">c.</abbr> 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats (Fm+1) is obtained by adding one [S] to the Fm cases and one [L] to the Fm−1 cases.<a href="#cite_note-12">[12]</a> <a href="Bharata_Muni" class="mw-redirect" title="Bharata Muni">Bharata Muni</a> also expresses knowledge of the sequence in the <a href="Natya_Shastra" title="Natya Shastra">Natya Shastra</a> (c. 100 BC–c. 350 AD).<a href="#cite_note-GlobalScience-3">[3]</a><a href="#cite_note-HistoriaMathematica-4">[4]</a>

However, the clearest exposition of the sequence arises in the work of <a href="Virahanka" title="Virahanka">Virahanka</a> (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):<a href="#cite_note-FOOTNOTELivio2003197-11">[11]</a>

<blockquote>Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].<a href="#cite_note-14">[a]</a></blockquote>

<a href="Hemachandra" title="Hemachandra">Hemachandra</a> (c. 1150) is credited with knowledge of the sequence as well,<a href="#cite_note-GlobalScience-3">[3]</a> writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."<a href="#cite_note-FOOTNOTELivio2003197–198-15">[14]</a><a href="#cite_note-16">[15]</a>

<div class="mw-heading mw-heading3"><h3 id="Europe">Europe</h3></div>

The Fibonacci sequence first appears in the book <a href="Liber_Abaci" title="Liber Abaci">Liber Abaci</a> (The Book of Calculation, 1202) by <a href="Fibonacci" title="Fibonacci">Fibonacci</a>,<a href="#cite_note-FOOTNOTESigler2002404–405-17">[16]</a><a href="#cite_note-18">[17]</a> where it is used to calculate the growth of rabbit populations.<a href="#cite_note-19">[18]</a> Fibonacci considers the growth of an idealized (<a href="Biology" title="Biology">biologically</a> unrealistic) <a href="Rabbit" title="Rabbit">rabbit</a> population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit <a href="Mathematical_problem" title="Mathematical problem">math problem</a>: how many pairs will there be in one year?

<ul><li>At the end of the first month, they mate, but there is still only 1 pair.</li>

<li>At the end of the second month they produce a new pair, so there are 2 pairs in the field.</li>

<li>At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.</li>

<li>At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.</li></ul>

At the end of the n-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2) plus the number of pairs alive last month (month n – 1). The number in the n-th month is the n-th Fibonacci number.<a href="#cite_note-20">[19]</a>

The name "Fibonacci sequence" was first used by the 19th-century number theorist <a href="%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>.<a href="#cite_note-21">[20]</a>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_golden_ratio">Relation to the golden ratio</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Golden_ratio" title="Golden ratio">Golden ratio</a></div>

<div class="mw-heading mw-heading3"><h3 id="Closed-form_expression">Closed-form expression </h3></div>

Like every <a href="Sequence" title="Sequence">sequence</a> defined by a homogeneous <a href="Linear_recurrence_with_constant_coefficients" title="Linear recurrence with constant coefficients">linear recurrence with constant coefficients</a>, the Fibonacci numbers have a <a href="Closed-form_expression" title="Closed-form expression">closed-form expression</a>.<a href="#cite_note-22">[21]</a> It has become known as Binet's formula, named after French mathematician <a href="Jacques_Philippe_Marie_Binet" title="Jacques Philippe Marie Binet">Jacques Philippe Marie Binet</a>, though it was already known by <a href="Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> and <a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>:<a href="#cite_note-23">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }</annotation>

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is the <a href="Golden_ratio" title="Golden ratio">golden ratio</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> is its <a href="Conjugate_(square_roots)" title="Conjugate (square roots)">conjugate</a>:<a href="#cite_note-FOOTNOTEBall2003156-24">[23]</a>

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<annotation encoding="application/x-tex">{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}</annotation>

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Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-\varphi ^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi =-\varphi ^{-1}}</annotation>

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</math><img src="./d966e5006d42590d61ad2416b47a178c783bfde6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.273ex; height:3.176ex;" alt="{\displaystyle \psi =-\varphi ^{-1}}" loading="lazy">, this formula can also be written as

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<annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}</annotation>

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To see the relation between the sequence and these constants,<a href="#cite_note-FOOTNOTEBall2003155–156-25">[24]</a> note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./afcaaabe7eb8ba6c20cca6efe1f46c8acdf13a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.676ex;" alt="{\textstyle x^{2}=x+1}" loading="lazy"> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}=x^{n-1}+x^{n-2},}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}=x^{n-1}+x^{n-2},}</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> satisfy the Fibonacci recursion. In other words,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}</annotation>

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It follows that for any values a and b, the sequence defined by

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<annotation encoding="application/x-tex">{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}</annotation>

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satisfies the same recurrence,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}</annotation>

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If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

<mrow>

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<mtd>

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<mtd>

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<mtr>

<mtd>

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}</annotation>

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which has solution

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}</annotation>

</semantics>

</math>

producing the required formula.

Taking the starting values U0 and U1 to be arbitrary constants, a more general solution is:

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}</annotation>

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</math>

where

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<mrow>

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<msub>

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Computation_by_rounding">Computation by rounding</h3></div>

Since

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<mfrac>

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<annotation encoding="application/x-tex">{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./2f36056259148225cdc34612e19c42887fb89055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.077ex; height:4.843ex;" alt="{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}" loading="lazy"> for all n ≥ 0, the number Fn is the closest <a href="Integer" title="Integer">integer</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}">

<mfrac>

<msup>

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</msup>

<msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}</annotation>

</semantics>

</math><img src="./67180162747d0e482d1bbb421cee2742f2e303c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:6.343ex;" alt="{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}" loading="lazy">. Therefore, it can be found by <a href="Rounding" title="Rounding">rounding</a>, using the nearest integer function:

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<annotation encoding="application/x-tex">{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}</annotation>

</semantics>

</math>

In fact, the rounding error quickly becomes very small as n grows, being less than 0.1 for n ≥ 4, and less than 0.01 for n ≥ 8. This formula is easily inverted to find an index of a Fibonacci number F:

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<annotation encoding="application/x-tex">{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}</annotation>

</semantics>

</math>

Instead using the <a href="Floor_function" class="mw-redirect" title="Floor function">floor function</a> gives the largest index of a Fibonacci number that is not greater than F:

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}">

<msub>

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<msub>

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</mrow>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}</annotation>

</semantics>

</math><img src="./010eea776ca1d745f74b0e4a1aefdc563315a3c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.225ex; height:3.176ex;" alt="{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(\varphi )=0.481211\ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln(\varphi )=0.481211\ldots }</annotation>

</semantics>

</math><img src="./bbe83479430c5a199b4657e1476f11e01f9bc99b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.262ex; height:2.843ex;" alt="{\displaystyle \ln(\varphi )=0.481211\ldots }" loading="lazy">,<a href="#cite_note-26">[25]</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(\varphi )=0.208987\ldots }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log _{10}(\varphi )=0.208987\ldots }</annotation>

</semantics>

</math><img src="./411e51d091b7b5f93fa086bb2bdfd2f72d2f27d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.17ex; height:2.843ex;" alt="{\displaystyle \log _{10}(\varphi )=0.208987\ldots }" loading="lazy">.<a href="#cite_note-27">[26]</a>

<div class="mw-heading mw-heading3"><h3 id="Magnitude">Magnitude</h3></div>

Since Fn is <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}/{\sqrt {5}}}">

<msup>

</mrow>

</msup>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}/{\sqrt {5}}}</annotation>

</semantics>

</math><img src="./e44feeb18cba5ae7d9cef89d5bc2c4b016b71161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.999ex; height:3.009ex;" alt="{\displaystyle \varphi ^{n}/{\sqrt {5}}}" loading="lazy">, the number of digits in Fn is asymptotic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}</annotation>

</semantics>

</math><img src="./c17e5414f184956bc08aa6f0447847e1eddbfbb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.876ex; height:2.676ex;" alt="{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}" loading="lazy">. As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.

More generally, in the <a href="Radix" title="Radix">base</a> b representation, the number of digits in Fn is asymptotic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}">

<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}</annotation>

</semantics>

</math><img src="./1f7f35205ae578d5571a57374c0f225b169a5554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.841ex; height:5.843ex;" alt="{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Limit_of_consecutive_quotients">Limit of consecutive quotients</h3></div>

<a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> observed that the ratio of consecutive Fibonacci numbers <a href="Convergent_sequence" class="mw-redirect" title="Convergent sequence">converges</a>. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy">⁠:<a href="#cite_note-28">[27]</a><a href="#cite_note-29">[28]</a>

</mrow>

</munder>

<mfrac>

<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}</annotation>

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</math>

This convergence holds regardless of the starting values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle U_{0}}</annotation>

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</math><img src="./52116e300c6199496d7b4a60d417ba34a4e569dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{1}}</annotation>

</semantics>

</math><img src="./bc9e7f892894bc50c32ce1b9f9a68a15562146ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{1}}" loading="lazy">, unless <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}=-U_{0}/\varphi }">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle U_{1}=-U_{0}/\varphi }</annotation>

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</math><img src="./3a6f0e3daace50ce6d41cd734bd07fe0e9a38572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.873ex; height:2.843ex;" alt="{\displaystyle U_{1}=-U_{0}/\varphi }" loading="lazy">. This can be verified using <a href="#Binet's_formula">Binet's formula</a>. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive elements in this sequence shows the same convergence towards the golden ratio.

In general, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}</annotation>

</semantics>

</math><img src="./57c7440ca49a11b48c02fdd2366e5f8b87036bda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.224ex; height:5.676ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}" loading="lazy">, because the ratios between consecutive Fibonacci numbers approaches <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Decomposition_of_powers">Decomposition of powers</h3></div>

Since the golden ratio satisfies the equation

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \varphi ^{2}=\varphi +1,}</annotation>

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this expression can be used to decompose higher powers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation>

</semantics>

</math><img src="./2bfefb926a7e3f934344edc2d382566ad9c37c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}" loading="lazy"> as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and 1. The resulting <a href="Recurrence_relation" title="Recurrence relation">recurrence relationships</a> yield Fibonacci numbers as the linear <a href="Coefficient" title="Coefficient">coefficients</a>:

<msup>

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</msup>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}</annotation>

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</math>

This equation can be <a href="Mathematical_proof" title="Mathematical proof">proved</a> by <a href="Mathematical_induction" title="Mathematical induction">induction</a> on n ≥ 1:

<mtr>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi ^{n+1}&=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi \\&=F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.\end{aligned}}}</annotation>

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</math>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-1/\varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi =-1/\varphi }</annotation>

</semantics>

</math><img src="./91b27868e4f91891f5d1fa260c9acce9018dd638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.265ex; height:2.843ex;" alt="{\displaystyle \psi =-1/\varphi }" loading="lazy">, it is also the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{2}=\psi +1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \psi ^{2}=\psi +1}</annotation>

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</math><img src="./6fff2f022bcd58d953fb30897d6ba19a94f6c8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.182ex; height:3.009ex;" alt="{\displaystyle \psi ^{2}=\psi +1}" loading="lazy"> and it is also the case that

<msup>

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</msup>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}</annotation>

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</math>

These expressions are also true for n < 1 if the Fibonacci sequence Fn is <a href="Generalizations_of_Fibonacci_numbers#Extension_to_negative_integers" title="Generalizations of Fibonacci numbers">extended to negative integers</a> using the Fibonacci rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}">

<msub>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}</annotation>

</semantics>

</math><img src="./5aa79f32a1e7651f9ad22a00a09646a0898d30ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.926ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Identification">Identification</h3></div>

Binet's formula provides a proof that a positive integer x is a Fibonacci number <a href="If_and_only_if" title="If and only if">if and only if</a> at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}+4}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 5x^{2}+4}</annotation>

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</math><img src="./63265a5787c1994492f335bf4957ff17524ca4e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 5x^{2}+4}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}-4}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 5x^{2}-4}</annotation>

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</math><img src="./fcba7f88ca4dd3784151c47c6107dd16124bc7c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 5x^{2}-4}" loading="lazy"> is a <a href="Square_number" title="Square number">perfect square</a>.<a href="#cite_note-30">[29]</a> This is because Binet's formula, which can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}">

<msub>

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<msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}</annotation>

</semantics>

</math><img src="./7f55805af793c7b658d259a3d00a98533e49ff72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.476ex; height:3.009ex;" alt="{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}" loading="lazy">, can be multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}\varphi ^{n}}">

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}\varphi ^{n}}</annotation>

</semantics>

</math><img src="./f439472ea2a40ccc0863e2c1c6a0280002493b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.837ex; height:3.009ex;" alt="{\displaystyle {\sqrt {5}}\varphi ^{n}}" loading="lazy"> and solved as a <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation>

</semantics>

</math><img src="./2bfefb926a7e3f934344edc2d382566ad9c37c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}" loading="lazy"> via the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a>:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{\!2}+4(-1)^{n}}}}{2}}.}</annotation>

</semantics>

</math>

Comparing this to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}</annotation>

</semantics>

</math><img src="./66791cc13a236899e54dc6135a7b787e74696bee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.138ex; height:3.009ex;" alt="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}" loading="lazy">, it follows that

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5{F_{n}}^{\!2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}</annotation>

</semantics>

</math></dd></dl>

In particular, the left-hand side is a perfect square.

<div class="mw-heading mw-heading2"><h2 id="Matrix_form">Matrix form</h2></div>

A 2-dimensional system of linear <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a> that describes the Fibonacci sequence is

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}F_{k+2}\\F_{k+1}\end{pmatrix}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{\begin{pmatrix}F_{k+1}\\F_{k}\end{pmatrix}}}</annotation>

</semantics>

</math>

alternatively denoted

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}</annotation>

</semantics>

</math>

which yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}</annotation>

</semantics>

</math><img src="./876b987b76e01bf64355805883dd71b3fbec01e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.151ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}" loading="lazy">. The <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> A are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}</annotation>

</semantics>

</math><img src="./bf5ab0819fbf64ef2942ee3e805000ef40f24849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.701ex; height:3.509ex;" alt="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}">

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}</annotation>

</semantics>

</math><img src="./2a24efb7dab0274af1074f2a7a09eabfaa80bec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.454ex; height:3.509ex;" alt="{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}" loading="lazy"> corresponding to the respective <a href="Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\mu }}={\begin{pmatrix}\varphi \\1\end{pmatrix}},\quad {\vec {\nu }}={\begin{pmatrix}-\varphi ^{-1}\\1\end{pmatrix}}.}</annotation>

</semantics>

</math>

As the initial value is

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{0}={\begin{pmatrix}1\\0\end{pmatrix}}={\frac {1}{\sqrt {5}}}{\vec {\mu }}\,-\,{\frac {1}{\sqrt {5}}}{\vec {\nu }},}</annotation>

</semantics>

</math>

it follows that the nth element is

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {F}}_{n}\ &={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\begin{pmatrix}\varphi \\1\end{pmatrix}}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{\begin{pmatrix}{c}-\varphi ^{-1}\\1\end{pmatrix}}.\end{aligned}}}</annotation>

</semantics>

</math>

From this, the nth element in the Fibonacci series may be read off directly as a <a href="Closed-form_expression" title="Closed-form expression">closed-form expression</a>:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}</annotation>

</semantics>

</math>

Equivalently, the same computation may be performed by <a href="Matrix_diagonalization" class="mw-redirect" title="Matrix diagonalization">diagonalization</a> of A through use of its <a href="Eigendecomposition" class="mw-redirect" title="Eigendecomposition">eigendecomposition</a>:

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\[3mu]A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}</annotation>

</semantics>

</math>

where

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}</annotation>

</semantics>

</math>

The closed-form expression for the nth element in the Fibonacci series is therefore given by

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}F_{n+1}\\F_{n}\end{pmatrix}}&=A^{n}{\begin{pmatrix}F_{1}\\F_{0}\end{pmatrix}}\ \\&=S\Lambda ^{n}S^{-1}{\begin{pmatrix}F_{1}\\F_{0}\end{pmatrix}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{\begin{pmatrix}F_{1}\\F_{0}\end{pmatrix}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{\begin{pmatrix}1\\0\end{pmatrix}},\end{aligned}}}</annotation>

</semantics>

</math>

which again yields

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}</annotation>

</semantics>

</math>

The matrix A has a <a href="Determinant" title="Determinant">determinant</a> of −1, and thus it is a 2 × 2 <a href="Unimodular_matrix" title="Unimodular matrix">unimodular matrix</a>.

This property can be understood in terms of the <a href="Continued_fraction" title="Continued fraction">continued fraction</a> representation for the golden ratio φ:

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}</annotation>

</semantics>

</math>

The <a href="Convergent_(continued_fraction)" class="mw-redirect" title="Convergent (continued fraction)">convergents</a> of the continued fraction for φ are ratios of successive Fibonacci numbers: φn = Fn+1 / Fn is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φn+1 = 1 + 1 / φn.<a href="#cite_note-31">[30]</a> The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}</annotation>

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For a given n, this matrix can be computed in O(log n) arithmetic operations, using the <a href="Exponentiation_by_squaring" title="Exponentiation by squaring">exponentiation by squaring</a> method.

Taking the determinant of both sides of this equation yields <a href="Cassini's_identity" class="mw-redirect" title="Cassini's identity">Cassini's identity</a>,

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<annotation encoding="application/x-tex">{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}</annotation>

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Moreover, since AnAm = An+m for any <a href="Square_matrix" title="Square matrix">square matrix</a> A, the following <a href="Identity_(mathematics)" title="Identity (mathematics)">identities</a> can be derived (they are obtained from two different coefficients of the <a href="Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a>, and one may easily deduce the second one from the first one by changing n into n + 1),

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\[3mu]F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}</annotation>

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In particular, with m = n,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\[6mu]F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\[3mu]&=(2F_{n-1}+F_{n})F_{n}\\[3mu]&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}</annotation>

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These last two identities provide a way to compute Fibonacci numbers <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursively</a> in O(log n) arithmetic operations. This matches the time for computing the n-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with <a href="Memoization" title="Memoization">memoization</a>).<a href="#cite_note-32">[31]</a>

<div class="mw-heading mw-heading2"><h2 id="Combinatorial_identities">Combinatorial identities</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Combinatorial_proofs">Combinatorial proofs</h3></div>

Most identities involving Fibonacci numbers can be proved using <a href="Combinatorial_proof" title="Combinatorial proof">combinatorial arguments</a> using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

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</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

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</math><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy">. This can be taken as the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

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</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> with the conventions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0}">

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</math><img src="./58ebe8b2d5551fb272cd4258940fe1e492592d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=0}" loading="lazy">, meaning no such sequence exists whose sum is −1, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}">

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</math><img src="./c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}" loading="lazy">, meaning the empty sequence "adds up" to 0. In the following, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{...}|}">

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</math><img src="./75f05f18aeb1ba679b1139aa940ce5b761017c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.008ex; height:2.843ex;" alt="{\displaystyle |{...}|}" loading="lazy"> is the <a href="Cardinality" title="Cardinality">cardinality</a> of a <a href="Set_(mathematics)" title="Set (mathematics)">set</a>:

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<annotation encoding="application/x-tex">{\displaystyle F_{0}=0=|\{\}|}</annotation>

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</math><img src="./11d5f5532a775ac0eff7962124e5498d6a3cc83a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.527ex; height:2.843ex;" alt="{\displaystyle F_{0}=0=|\{\}|}" loading="lazy"></dd>

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</math><img src="./50caafd76b616284b322433b7cfb86999e1e65ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.336ex; height:2.843ex;" alt="{\displaystyle F_{1}=1=|\{()\}|}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle F_{2}=1=|\{(1)\}|}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}</annotation>

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In this manner the recurrence relation

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may be understood by dividing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

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Excluding the first element, the remaining terms in each sequence sum to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-2}">

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In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the n-th is equal to the (n + 2)-th Fibonacci number minus 1.<a href="#cite_note-FOOTNOTELucas18914-33">[32]</a> In symbols:

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This may be seen by dividing all sequences summing to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

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</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> based on the location of the first 2. Specifically, each set consists of those sequences that start <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2,...),(1,2,...),...,}">

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Following the same logic as before, by summing the cardinality of each set we see that

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<annotation encoding="application/x-tex">{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}</annotation>

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</math><img src="./f3d7499d2f1cb8b0d34907c42161e9788bf1eda8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.272ex; height:2.843ex;" alt="{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}" loading="lazy"></dd></dl>

... where the last two terms have the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}">

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</math><img src="./c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}" loading="lazy">. From this it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}</annotation>

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A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}</annotation>

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In words, the sum of the first Fibonacci numbers with <a href="Parity_(mathematics)" title="Parity (mathematics)">odd</a> index up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2n-1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{2n-1}}</annotation>

</semantics>

</math><img src="./da1f7e7003160cda89bfedb51e7971f700d1457c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.636ex; height:2.509ex;" alt="{\displaystyle F_{2n-1}}" loading="lazy"> is the (2n)-th Fibonacci number, and the sum of the first Fibonacci numbers with <a href="Parity_(mathematics)" title="Parity (mathematics)">even</a> index up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{2n}}</annotation>

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</math><img src="./191c2d8c9a1430c1d33704b6240aa12881382496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.535ex; height:2.509ex;" alt="{\displaystyle F_{2n}}" loading="lazy"> is the (2n + 1)-th Fibonacci number minus 1.<a href="#cite_note-34">[33]</a>

A different trick may be used to prove

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}</annotation>

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or in words, the sum of the squares of the first Fibonacci numbers up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

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</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> is the product of the n-th and (n + 1)-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\times F_{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{n}\times F_{n+1}}</annotation>

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</math><img src="./8cb44dffb7e5fa0286f5dd41eb1e9a69b7a00afd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.367ex; height:2.509ex;" alt="{\displaystyle F_{n}\times F_{n+1}}" loading="lazy"> and decompose it into squares of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n},F_{n-1},...,F_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{n},F_{n-1},...,F_{1}}</annotation>

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</math><img src="./96ffdb7c33371dc43890e95708d85ff6ddcfaba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.279ex; height:2.509ex;" alt="{\displaystyle F_{n},F_{n-1},...,F_{1}}" loading="lazy">; from this the identity follows by comparing areas:

<div class="mw-heading mw-heading3"><h3 id="Induction_proofs">Induction proofs</h3></div>

Fibonacci identities often can be easily proved using <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>.

For example, reconsider

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}</annotation>

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Adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{n+1}}</annotation>

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</math><img src="./7bfbe34f204a6b7b01dd49571e6b287c2bdf7735.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.814ex; height:2.509ex;" alt="{\displaystyle F_{n+1}}" loading="lazy"> to both sides gives

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}</annotation>

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</math><img src="./4f2641e2f9bbfb8cbb364883ae5a5f1751d69ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.259ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}" loading="lazy"></dd></dl>

and so we have the formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}</annotation>

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</math>

Similarly, add <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{n+1}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {F_{n+1}}^{2}}</annotation>

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</math><img src="./7197da79ba73a67b86f387fe8ffa56edab7a87cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.868ex; height:3.009ex;" alt="{\displaystyle {F_{n+1}}^{2}}" loading="lazy"> to both sides of

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}</annotation>

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to give

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Binet_formula_proofs">Binet formula proofs</h3></div>

The Binet formula is

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}</annotation>

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This can be used to prove Fibonacci identities.

For example, to prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}</annotation>

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</math><img src="./c6c2dfcb3f4762e5c4f8b5323583c0862d8b4a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.95ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}" loading="lazy">

note that the left hand side multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}}">

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}}</annotation>

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</math><img src="./2b78ccdb7e18e02d4fc567c66aac99bf524acb5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {5}}}" loading="lazy"> becomes

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}</annotation>

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</math>

as required, using the facts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi \psi =-1}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \varphi \psi =-1}</annotation>

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</math><img src="./b9ed2751085d1c14679f552b2bf4b4e434a38fb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.102ex; height:2.676ex;" alt="{\textstyle \varphi \psi =-1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi -\psi ={\sqrt {5}}}">

<msqrt>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi -\psi ={\sqrt {5}}}</annotation>

</semantics>

</math><img src="./6ef489f2b58745f57aadfd0a1bc704fdc4eeaa7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.07ex; height:3.176ex;" alt="{\textstyle \varphi -\psi ={\sqrt {5}}}" loading="lazy"> to simplify the equations.

<div class="mw-heading mw-heading2"><h2 id="Other_identities">Other identities</h2></div>

Numerous other identities can be derived using various methods. Here are some of them:<a href="#cite_note-MathWorld-35">[34]</a>

<div class="mw-heading mw-heading3"><h3 id="Cassini's_and_Catalan's_identities">Cassini's and Catalan's identities</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cassini_and_Catalan_identities" title="Cassini and Catalan identities">Cassini and Catalan identities</a></div>

Cassini's identity states that

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<annotation encoding="application/x-tex">{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}</annotation>

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</math>

Catalan's identity is a generalization:

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<annotation encoding="application/x-tex">{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}</annotation>

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</math>

<div class="mw-heading mw-heading3"><h3 id="d'Ocagne's_identity">d'Ocagne's identity</h3></div>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}</annotation>

</semantics>

</math>

where Ln is the n-th <a href="Lucas_number" title="Lucas number">Lucas number</a>. The last is an identity for doubling n; other identities of this type are

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<annotation encoding="application/x-tex">{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}</annotation>

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</math>

by Cassini's identity.

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<annotation encoding="application/x-tex">{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}</annotation>

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</math>

These can be found experimentally using <a href="Lattice_reduction" title="Lattice reduction">lattice reduction</a>, and are useful in setting up the <a href="Special_number_field_sieve" title="Special number field sieve">special number field sieve</a> to <a href="Factorization" title="Factorization">factorize</a> a Fibonacci number.

More generally,<a href="#cite_note-MathWorld-35">[34]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{\binom {k}{i}}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}</annotation>

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or alternatively

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<annotation encoding="application/x-tex">{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{\binom {k}{i}}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}</annotation>

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</math>

Putting k = 2 in this formula, one gets again the formulas of the end of above section <a href="#Matrix_form">Matrix form</a>.

<div class="mw-heading mw-heading2"><h2 id="Generating_function">Generating function</h2></div>

The <a href="Generating_function" title="Generating function">generating function</a> of the Fibonacci sequence is the <a href="Power_series" title="Power series">power series</a>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\cdots .}</annotation>

</semantics>

</math>

This series is convergent for any <a href="Complex_number" title="Complex number">complex number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1/\varphi \approx 0.618,}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |z|<1/\varphi \approx 0.618,}</annotation>

</semantics>

</math><img src="./8ce62aaa0b60075574b8a9506e1dd2f0fdf4392e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.367ex; height:2.843ex;" alt="{\displaystyle |z|<1/\varphi \approx 0.618,}" loading="lazy"> and its sum has a simple closed form:<a href="#cite_note-36">[35]</a>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}</annotation>

</semantics>

</math>

This can be proved by multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (1-z-z^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (1-z-z^{2})}</annotation>

</semantics>

</math><img src="./ca577c4a9a39370f2542f0db98c7cdba2b471414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.885ex; height:3.009ex;" alt="{\textstyle (1-z-z^{2})}" loading="lazy">:

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}</annotation>

</semantics>

</math>

where all terms involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{k}}</annotation>

</semantics>

</math><img src="./152bec16756f5bbf6fa6038d8f8d022923cb7b06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.179ex; height:2.676ex;" alt="{\displaystyle z^{k}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation>

</semantics>

</math><img src="./c797a67c0a51167d373c013a9a020f4568a11754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}" loading="lazy"> cancel out because of the defining Fibonacci recurrence relation.

The <a href="Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a> is given by

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}</annotation>

</semantics>

</math><img src="./f7b287380393d68ac21a9b14367d36530b4685c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.895ex; height:3.509ex;" alt="{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}" loading="lazy"> is the golden ratio and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}</annotation>

</semantics>

</math><img src="./57a0e2b9cc155c9737fea8786161f3422f432849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.888ex; height:3.509ex;" alt="{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}" loading="lazy"> is its <a href="Conjugate_(square_roots)" title="Conjugate (square roots)">conjugate</a>.

The related function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z\mapsto -s\left(-1/z\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle z\mapsto -s\left(-1/z\right)}</annotation>

</semantics>

</math><img src="./d91eb017da4a8e875ba62889165897bd8ee40ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.018ex; height:2.843ex;" alt="{\textstyle z\mapsto -s\left(-1/z\right)}" loading="lazy"> is the generating function for the <a href="Negafibonacci" class="mw-redirect" title="Negafibonacci">negafibonacci</a> numbers, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)}</annotation>

</semantics>

</math><img src="./7bf6ea9e3135132b01ef408198af8a1470edfe15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle s(z)}" loading="lazy"> satisfies the <a href="Functional_equation" title="Functional equation">functional equation</a>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}</annotation>

</semantics>

</math>

Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(z)}</annotation>

</semantics>

</math><img src="./7bf6ea9e3135132b01ef408198af8a1470edfe15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle s(z)}" loading="lazy">. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}</annotation>

</semantics>

</math><img src="./71b1cfea70bdd62a2e447aef380cdc233095ade8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.775ex; height:5.176ex;" alt="{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Reciprocal_sums">Reciprocal sums</h2></div>

Infinite sums over <a href="Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> Fibonacci numbers can sometimes be evaluated in terms of <a href="Theta_function" title="Theta function">theta functions</a>. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}</annotation>

</semantics>

</math>

and the sum of squared reciprocal Fibonacci numbers as

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}</annotation>

</semantics>

</math>

If we add 1 to each Fibonacci number in the first sum, there is also the closed form

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}</annotation>

</semantics>

</math>

and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the <a href="Golden_ratio" title="Golden ratio">golden ratio</a>,

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}</annotation>

</semantics>

</math>

The sum of all even-indexed reciprocal Fibonacci numbers is<a href="#cite_note-37">[36]</a>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}</annotation>

</semantics>

</math>

with the <a href="Lambert_series" title="Lambert series">Lambert series</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}">

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}</annotation>

</semantics>

</math><img src="./136916fcbd27ca5379d3b54f4e05b7609fa5c6e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.444ex; height:5.009ex;" alt="{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}" loading="lazy"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}</annotation>

</semantics>

</math><img src="./745a38c05a7430bde48a6bb6d95b6bd7cf305bf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.979ex; height:6.176ex;" alt="{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}" loading="lazy">

So the <a href="Reciprocal_Fibonacci_constant" title="Reciprocal Fibonacci constant">reciprocal Fibonacci constant</a> is<a href="#cite_note-38">[37]</a>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }</annotation>

</semantics>

</math>

Moreover, this number has been proved <a href="Irrational_number" title="Irrational number">irrational</a> by Richard André-Jeannin.<a href="#cite_note-39">[38]</a>

Millin's series gives the identity<a href="#cite_note-40">[39]</a>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}</annotation>

</semantics>

</math>

which follows from the closed form for its partial sums as N tends to infinity:

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Primes_and_divisibility">Primes and divisibility</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Divisibility_properties">Divisibility properties</h3></div>

Every third number of the sequence is even (a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{3}=2}</annotation>

</semantics>

</math><img src="./1caf4ec3fa730d75d5508487f92ebe0a64307a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{3}=2}" loading="lazy">) and, more generally, every k-th number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a <a href="Divisibility_sequence" title="Divisibility sequence">divisibility sequence</a>. In fact, the Fibonacci sequence satisfies the stronger divisibility property<a href="#cite_note-41">[40]</a><a href="#cite_note-42">[41]</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}</annotation>

</semantics>

</math>

where gcd is the <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> function. (This relation is different if a different indexing convention is used, such as the one that starts the sequence with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=1}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{0}=1}</annotation>

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In particular, any three consecutive Fibonacci numbers are pairwise <a href="Coprime_integers" title="Coprime integers">coprime</a> because both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}">

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</math><img src="./c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=1}">

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</math><img src="./fdf100a6879452aff05a6027ad4f36029f360dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{2}=1}" loading="lazy">. That is,

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<annotation encoding="application/x-tex">{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}</annotation>

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for every n.

Every <a href="Prime_number" title="Prime number">prime number</a> p divides a Fibonacci number that can be determined by the value of p <a href="Modular_arithmetic" title="Modular arithmetic">modulo</a> 5. If p is congruent to 1 or 4 modulo 5, then p divides Fp−1, and if p is congruent to 2 or 3 modulo 5, then, p divides Fp+1. The remaining case is that p = 5, and in this case p divides Fp.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}</annotation>

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These cases can be combined into a single, non-<a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> formula, using the <a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a>:<a href="#cite_note-43">[42]</a>

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<annotation encoding="application/x-tex">{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Primality_testing">Primality testing</h3></div>

The above formula can be used as a <a href="Primality_test" title="Primality test">primality test</a> in the sense that if

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where the Legendre symbol has been replaced by the <a href="Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a>, then this is evidence that n is a prime, and if it fails to hold, then n is definitely not a prime. If n is <a href="Composite_number" title="Composite number">composite</a> and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large – say a 500-<a href="Bit" title="Bit">bit</a> number – then we can calculate Fm (mod n) efficiently using the matrix form. Thus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}</annotation>

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Here the matrix power Am is calculated using <a href="Modular_exponentiation" title="Modular exponentiation">modular exponentiation</a>, which can be <a href="Modular_exponentiation#Matrices" title="Modular exponentiation">adapted to matrices</a>.<a href="#cite_note-44">[43]</a>

<div class="mw-heading mw-heading3"><h3 id="Fibonacci_primes">Fibonacci primes</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fibonacci_prime" title="Fibonacci prime">Fibonacci prime</a></div>

A Fibonacci prime is a Fibonacci number that is <a href="Prime_number" title="Prime number">prime</a>. The first few are:<a href="#cite_note-45">[44]</a>

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.<a href="#cite_note-46">[45]</a>

Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are <a href="Arbitrarily_large" title="Arbitrarily large">arbitrarily long</a> runs of <a href="Composite_number" title="Composite number">composite numbers</a>, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.<a href="#cite_note-47">[46]</a>

The only nontrivial <a href="Square_number" title="Square number">square</a> Fibonacci number is 144.<a href="#cite_note-48">[47]</a> Attila Pethő proved in 2001 that there is only a finite number of <a href="Perfect_power" title="Perfect power">perfect power</a> Fibonacci numbers.<a href="#cite_note-49">[48]</a> In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.<a href="#cite_note-50">[49]</a>

1, 3, 21, and 55 are the only <a href="Triangular_number" title="Triangular number">triangular</a> Fibonacci numbers, which was <a href="Conjecture" title="Conjecture">conjectured</a> by <a href="Verner_Emil_Hoggatt_Jr." title="Verner Emil Hoggatt Jr.">Vern Hoggatt</a> and proved by Luo Ming.<a href="#cite_note-51">[50]</a>

No Fibonacci number can be a <a href="Perfect_number" title="Perfect number">perfect number</a>.<a href="#cite_note-Luca2000-52">[51]</a> More generally, no Fibonacci number other than 1 can be <a href="Multiply_perfect_number" title="Multiply perfect number">multiply perfect</a>,<a href="#cite_note-BGLLHT2011-53">[52]</a> and no ratio of two Fibonacci numbers can be perfect.<a href="#cite_note-LucaMH2010-54">[53]</a>

<div class="mw-heading mw-heading3"><h3 id="Prime_divisors">Prime divisors</h3></div>

With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (<a href="Carmichael's_theorem" title="Carmichael's theorem">Carmichael's theorem</a>).<a href="#cite_note-55">[54]</a> As a result, 8 and 144 (F6 and F12) are the only Fibonacci numbers that are the product of other Fibonacci numbers.<a href="#cite_note-56">[55]</a>

The divisibility of Fibonacci numbers by a prime p is related to the <a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}</annotation>

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If p is a prime number then

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<annotation encoding="application/x-tex">{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}</annotation>

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</math><a href="#cite_note-57">[56]</a><a href="#cite_note-FOOTNOTELemmermeyer200073–74ex._2.25–28-58">[57]</a>

For example,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}</annotation>

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It is not known whether there exists a prime p such that

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<annotation encoding="application/x-tex">{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}</annotation>

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Such primes (if there are any) would be called <a href="Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun primes</a>.

Also, if p ≠ 5 is an odd prime number then:<a href="#cite_note-FOOTNOTELemmermeyer200073–74ex._2.28-59">[58]</a>

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<annotation encoding="application/x-tex">{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}</annotation>

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Example 1. p = 7, in this case p ≡ 3 (mod 4) and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}</annotation>

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Example 2. p = 11, in this case p ≡ 3 (mod 4) and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}</annotation>

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Example 3. p = 13, in this case p ≡ 1 (mod 4) and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}</annotation>

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Example 4. p = 29, in this case p ≡ 1 (mod 4) and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}</annotation>

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</math>

For odd n, all odd prime divisors of Fn are congruent to 1 modulo 4, implying that all odd divisors of Fn (as the products of odd prime divisors) are congruent to 1 modulo 4.<a href="#cite_note-FOOTNOTELemmermeyer200073ex._2.27-60">[59]</a>

For example,

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<annotation encoding="application/x-tex">{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}</annotation>

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</math>

All known factors of Fibonacci numbers F(i) for all i < 50000 are collected at the relevant repositories.<a href="#cite_note-61">[60]</a><a href="#cite_note-62">[61]</a>

<div class="mw-heading mw-heading3"><h3 id="Periodicity_modulo_n">Periodicity modulo n</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Pisano_period" title="Pisano period">Pisano period</a></div>

If the members of the Fibonacci sequence are taken mod n, the resulting sequence is <a href="Periodic_sequence" title="Periodic sequence">periodic</a> with period at most 6n.<a href="#cite_note-63">[62]</a> The lengths of the periods for various n form the so-called <a href="Pisano_period" title="Pisano period">Pisano periods</a>.<a href="#cite_note-64">[63]</a> Determining a general formula for the Pisano periods is an <a href="Open_problem" title="Open problem">open problem</a>, which includes as a subproblem a special instance of the problem of finding the <a href="Multiplicative_order" title="Multiplicative order">multiplicative order</a> of a <a href="Modular_arithmetic" title="Modular arithmetic">modular integer</a> or of an element in a <a href="Finite_field" title="Finite field">finite field</a>. However, for any particular n, the Pisano period may be found as an instance of <a href="Cycle_detection" title="Cycle detection">cycle detection</a>.

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Generalizations_of_Fibonacci_numbers" title="Generalizations of Fibonacci numbers">Generalizations of Fibonacci numbers</a></div>

The Fibonacci sequence is one of the simplest and earliest known sequences defined by a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>, and specifically by a linear <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a <a href="Linear_recurrence_with_constant_coefficients" title="Linear recurrence with constant coefficients">homogeneous linear difference equation with constant coefficients</a>.

Some specific examples that are close, in some sense, to the Fibonacci sequence include:

<ul><li>Generalizing the index to negative integers to produce the <a href="Negafibonacci" class="mw-redirect" title="Negafibonacci">negafibonacci</a> numbers.</li>

<li>Generalizing the index to <a href="Real_number" title="Real number">real numbers</a> using a modification of Binet's formula.<a href="#cite_note-MathWorld-35">[34]</a></li>

<li>Starting with other integers. <a href="Lucas_number" title="Lucas number">Lucas numbers</a> have L1 = 1, L2 = 3, and Ln = Ln−1 + Ln−2. <a href="Primefree_sequence" title="Primefree sequence">Primefree sequences</a> use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.</li>

<li>Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The <a href="Pell_number" title="Pell number">Pell numbers</a> have Pn = 2Pn−1 + Pn−2. If the coefficient of the preceding value is assigned a variable value x, the result is the sequence of <a href="Fibonacci_polynomials" title="Fibonacci polynomials">Fibonacci polynomials</a>.</li>

<li>Not adding the immediately preceding numbers. The <a href="Padovan_sequence" title="Padovan sequence">Padovan sequence</a> and <a href="Perrin_number" title="Perrin number">Perrin numbers</a> have P(n) = P(n − 2) + P(n − 3).</li>

<li>Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers.<a href="#cite_note-65">[64]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3></div>

The Fibonacci numbers occur as the sums of <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> in the "shallow" diagonals of <a href="Pascal's_triangle" title="Pascal's triangle">Pascal's triangle</a>:<a href="#cite_note-FOOTNOTELucas18917-66">[65]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}</annotation>

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</math>

This can be proved by expanding the generating function

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<annotation encoding="application/x-tex">{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}</annotation>

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</math>

and collecting like terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation>

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</math><img src="./150d38e238991bc4d0689ffc9d2a852547d2658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" loading="lazy">.

To see how the formula is used, we can arrange the sums by the number of terms present:

<td>5

</td>

<td>= 1+1+1+1+1

</td></tr>

<tr>

<td>

</td>

<td>= 2+1+1+1

</td>

<td>= 1+2+1+1

</td>

<td>= 1+1+2+1

</td>

<td>= 1+1+1+2

</td></tr>

<tr>

<td>

</td>

<td>= 2+2+1

</td>

<td>= 2+1+2

</td>

<td>= 1+2+2

</td></tr></tbody></table></dd></dl>

which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}</annotation>

</semantics>

</math><img src="./a3b75215ce19fbf8985ab2740df1ea50bebbf3db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.536ex; height:3.509ex;" alt="{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}" loading="lazy">, where we are choosing the positions of k twos from n−k−1 terms.

These numbers also give the solution to certain enumerative problems,<a href="#cite_note-67">[66]</a> the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called <a href="Composition_(combinatorics)#Number_of_compositions" title="Composition (combinatorics)">compositions</a>); there are Fn+1 ways to do this (equivalently, it's also the number of <a href="Domino_tiling" title="Domino tiling">domino tilings</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 2\times n}</annotation>

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</math><img src="./d5155392314bbbc3e6304ec9dd8f7f633317fb87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 2\times n}" loading="lazy"> rectangle). For example, there are F5+1 = F6 = 8 ways one can climb a staircase of 5 steps, taking one or two steps at a time:

<td>5

</td>

<td>= 1+1+1+1+1

</td>

<td>= 2+1+1+1

</td>

<td>= 1+2+1+1

</td>

<td>= 1+1+2+1

</td>

<td>= 2+2+1

</td></tr>

<tr>

<td>

</td>

<td>= 1+1+1+2

</td>

<td>= 2+1+2

</td>

<td>= 1+2+2

</td></tr></tbody></table></dd></dl>

The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied <a href="Recursion" title="Recursion">recursively</a> until a single step, of which there is only one way to climb.

The Fibonacci numbers can be found in different ways among the set of <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> <a href="String_(computer_science)" title="String (computer science)">strings</a>, or equivalently, among the <a href="Subset" title="Subset">subsets</a> of a given set.

<ul><li>The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of <a href="Fibbinary_number" title="Fibbinary number">Fibbinary numbers</a>. Equivalently, Fn+2 is the number of subsets S of {1, ..., n} without consecutive integers, that is, those S for which {i, i + 1} ⊈ S for every i. A <a href="Bijection" title="Bijection">bijection</a> with the sums to n+1 is to replace 1 with 0 and 2 with 10, and drop the last zero.</li>

<li>The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S of {1, ..., n} without an odd number of consecutive integers is Fn+1. A bijection with the sums to n is to replace 1 with 0 and 2 with 11.</li>

<li>The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.</li>

<li><a href="Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> was able to show that the Fibonacci numbers can be defined by a <a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a>, which led to <a href="Matiyasevich's_theorem" class="mw-redirect" title="Matiyasevich's theorem">his solving</a> <a href="Hilbert's_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>.<a href="#cite_note-68">[67]</a></li>

<li>The Fibonacci numbers are also an example of a <a href="Complete_sequence" title="Complete sequence">complete sequence</a>. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.</li>

<li>Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as <a href="Zeckendorf's_theorem" title="Zeckendorf's theorem">Zeckendorf's theorem</a>, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its <a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci coding</a>.</li>

<li>Starting with 5, every second Fibonacci number is the length of the <a href="Hypotenuse" title="Hypotenuse">hypotenuse</a> of a <a href="Right_triangle" title="Right triangle">right triangle</a> with integer sides, or in other words, the largest number in a <a href="Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a>, obtained from the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}">

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<annotation encoding="application/x-tex">{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}</annotation>

</semantics>

</math> The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding triangle.<a href="#cite_note-69">[68]</a></li>

<li>The <a href="Fibonacci_cube" title="Fibonacci cube">Fibonacci cube</a> is an <a href="Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graph</a> with a Fibonacci number of nodes that has been proposed as a <a href="Network_topology" title="Network topology">network topology</a> for <a href="Parallel_computing" title="Parallel computing">parallel computing</a>.</li>

<li>Fibonacci numbers appear in the <a href="Ring_lemma" title="Ring lemma">ring lemma</a>, used to prove connections between the <a href="Circle_packing_theorem" title="Circle packing theorem">circle packing theorem</a> and <a href="Conformal_map" title="Conformal map">conformal maps</a>.<a href="#cite_note-70">[69]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3></div>

<ul><li>The Fibonacci numbers are important in <a href="Analysis_of_algorithms" title="Analysis of algorithms">computational run-time analysis</a> of <a href="Euclidean_algorithm" title="Euclidean algorithm">Euclid's algorithm</a> to determine the <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.<a href="#cite_note-71">[70]</a></li>

<li>Fibonacci numbers are used in a polyphase version of the <a href="Merge_sort" title="Merge sort">merge sort</a> algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the <a href="Polyphase_merge_sort" title="Polyphase merge sort">polyphase merge sort</a> was described in <a href="The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>.</li>

<li>A Fibonacci tree is a <a href="Binary_tree" title="Binary tree">binary tree</a> whose child trees (recursively) differ in <a href="Tree_height" class="mw-redirect" title="Tree height">height</a> by exactly 1. So it is an <a href="AVL_tree" title="AVL tree">AVL tree</a>, and one with the fewest nodes for a given height—the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.<a href="#cite_note-72">[71]</a></li>

<li>Fibonacci numbers are used by some <a href="Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a>.</li>

<li>Fibonacci numbers arise in the analysis of the <a href="Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> data structure.</li>

<li>A one-dimensional optimization method, called the <a href="Fibonacci_search_technique" title="Fibonacci search technique">Fibonacci search technique</a>, uses Fibonacci numbers.<a href="#cite_note-73">[72]</a></li>

<li>The Fibonacci number series is used for optional <a href="Lossy_compression" title="Lossy compression">lossy compression</a> in the <a href="Interchange_File_Format" title="Interchange File Format">IFF</a> <a href="8SVX" title="8SVX">8SVX</a> audio file format used on <a href="Amiga" title="Amiga">Amiga</a> computers. The number series <a href="Companding" title="Companding">compands</a> the original audio wave similar to logarithmic methods such as <a href="%CE%9C-law" class="mw-redirect" title="Μ-law">μ-law</a>.<a href="#cite_note-74">[73]</a><a href="#cite_note-75">[74]</a></li>

<li>Some Agile teams use a modified series called the "Modified Fibonacci Series" in <a href="Planning_poker" title="Planning poker">planning poker</a>, as an estimation tool. Planning Poker is a formal part of the <a href="Scaled_agile_framework" title="Scaled agile framework">Scaled Agile Framework</a>.<a href="#cite_note-76">[75]</a></li>

<li><a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci coding</a></li>

<li><a href="Negafibonacci_coding" title="Negafibonacci coding">Negafibonacci coding</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Nature">Nature</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Golden_ratio#Nature" title="Golden ratio">Golden ratio § Nature</a></div>

Fibonacci sequences appear in biological settings,<a href="#cite_note-77">[76]</a> such as branching in trees, <a href="Phyllotaxis" title="Phyllotaxis">arrangement of leaves on a stem</a>, the fruitlets of a <a href="Pineapple" title="Pineapple">pineapple</a>,<a href="#cite_note-78">[77]</a> the flowering of <a href="Artichoke" title="Artichoke">artichoke</a>, the arrangement of a <a href="Pine_cone" class="mw-redirect" title="Pine cone">pine cone</a>,<a href="#cite_note-79">[78]</a> and the family tree of <a href="Honeybee" class="mw-redirect" title="Honeybee">honeybees</a>.<a href="#cite_note-80">[79]</a><a href="#cite_note-81">[80]</a> <a href="Kepler" class="mw-redirect" title="Kepler">Kepler</a> pointed out the presence of the Fibonacci sequence in nature, using it to explain the (<a href="Golden_ratio" title="Golden ratio">golden ratio</a>-related) <a href="Pentagon" title="Pentagon">pentagonal</a> form of some flowers.<a href="#cite_note-FOOTNOTELivio2003110-82">[81]</a> Field <a href="Leucanthemum_vulgare" title="Leucanthemum vulgare">daisies</a> most often have petals in counts of Fibonacci numbers.<a href="#cite_note-FOOTNOTELivio2003112–13-83">[82]</a> In 1830, <a href="Karl_Friedrich_Schimper" title="Karl Friedrich Schimper">Karl Friedrich Schimper</a> and <a href="Alexander_Braun" title="Alexander Braun">Alexander Braun</a> discovered that the <a href="Parastichy" title="Parastichy">parastichies</a> (spiral <a href="Phyllotaxis" title="Phyllotaxis">phyllotaxis</a>) of plants were frequently expressed as fractions involving Fibonacci numbers.<a href="#cite_note-84">[83]</a>

<a href="Przemys%C5%82aw_Prusinkiewicz" title="Przemysław Prusinkiewicz">Przemysław Prusinkiewicz</a> advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on <a href="Free_group" title="Free group">free groups</a>, specifically as certain <a href="L-system" title="L-system">Lindenmayer grammars</a>.<a href="#cite_note-85">[84]</a>

A model for the pattern of <a href="Floret" class="mw-redirect" title="Floret">florets</a> in the head of a <a href="Sunflower" class="mw-redirect" title="Sunflower">sunflower</a> was proposed by Helmut Vogel in 1979.<a href="#cite_note-86">[85]</a> This has the form

<mfrac>

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</mfrac>

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<msqrt>

</msqrt>

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<annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}</annotation>

</semantics>

</math>

where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on <a href="Fermat's_spiral" title="Fermat's spiral">Fermat's spiral</a>. The divergence <a href="Angle" title="Angle">angle</a>, approximately 137.51°, is the <a href="Golden_angle" title="Golden angle">golden angle</a>, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F( j):F( j + 1), the nearest neighbors of floret number n are those at n ± F( j) for some index j, which depends on r, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,<a href="#cite_note-FOOTNOTELivio2003112-87">[86]</a> typically counted by the outermost range of radii.<a href="#cite_note-88">[87]</a>

Fibonacci numbers also appear in the ancestral pedigrees of <a href="Bee" title="Bee">bees</a> (which are <a href="Haplodiploid" class="mw-redirect" title="Haplodiploid">haplodiploids</a>), according to the following rules:

<ul><li>If an egg is laid but not fertilized, it produces a male (or <a href="Drone_(bee)" title="Drone (bee)">drone bee</a> in honeybees).</li>

<li>If, however, an egg is fertilized, it produces a female.</li></ul>

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, Fn, is the number of female ancestors, which is Fn−1, plus the number of male ancestors, which is Fn−2.<a href="#cite_note-89">[88]</a><a href="#cite_note-90">[89]</a> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

It has similarly been noticed that the number of possible ancestors on the human <a href="X_chromosome" title="X chromosome">X chromosome</a> inheritance line at a given ancestral generation also follows the Fibonacci sequence.<a href="#cite_note-xcs-91">[90]</a> A male individual has an X chromosome, which he received from his mother, and a <a href="Y_chromosome" title="Y chromosome">Y chromosome</a>, which he received from his father. The male counts as the "origin" of his own X chromosome (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}">

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation>

</semantics>

</math><img src="./c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}" loading="lazy">), and at his parents' generation, his X chromosome came from a single parent (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{2}=1}</annotation>

</semantics>

</math><img src="./fdf100a6879452aff05a6027ad4f36029f360dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{2}=1}" loading="lazy">). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{3}=2}</annotation>

</semantics>

</math><img src="./1caf4ec3fa730d75d5508487f92ebe0a64307a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{3}=2}" loading="lazy">). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}=3}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{4}=3}</annotation>

</semantics>

</math><img src="./3fcb7ddc635c556db90a8c277ce6b45e6d4aa185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{4}=3}" loading="lazy">). Five great-great-grandparents contributed to the male descendant's X chromosome (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}=5}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{5}=5}</annotation>

</semantics>

</math><img src="./767c16dde6476b8fd6618435d70266a3747e426f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{5}=5}" loading="lazy">), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a <a href="Founder_effect" title="Founder effect">population founder</a> appears on all lines of the genealogy.)

<div class="mw-heading mw-heading3"><h3 id="Other">Other</h3></div>

<ul><li>In <a href="Optics" title="Optics">optics</a>, when a beam of light shines at an angle through two stacked transparent plates of different materials of different <a href="Refractive_index" title="Refractive index">refractive indexes</a>, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have k reflections, for k > 1, is the k-th Fibonacci number. (However, when k = 1, there are three reflection paths, not two, one for each of the three surfaces.)<a href="#cite_note-FOOTNOTELivio200398–99-92">[91]</a></li>

<li><a href="Fibonacci_retracement" title="Fibonacci retracement">Fibonacci retracement</a> levels are widely used in <a href="Technical_analysis" title="Technical analysis">technical analysis</a> for financial market trading.</li>

<li>Since the <a href="Conversion_of_units" title="Conversion of units">conversion</a> factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a <a href="Radix" title="Radix">radix</a> 2 number <a href="Processor_register" title="Processor register">register</a> in <a href="Golden_ratio_base" title="Golden ratio base">golden ratio base</a> φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<a href="#cite_note-93">[92]</a></li>

<li>The measured values of voltages and currents in the infinite resistor chain circuit (also called the <a href="Resistor_ladder" title="Resistor ladder">resistor ladder</a> or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.<a href="#cite_note-94">[93]</a></li>

<li>Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of <a href="Economics" title="Economics">economics</a>.<a href="#cite_note-Brasch_et_al._2012-95">[94]</a> In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.</li>

<li><a href="Mario_Merz" title="Mario Merz">Mario Merz</a> included the Fibonacci sequence in some of his artworks beginning in 1970.<a href="#cite_note-FOOTNOTELivio2003176-96">[95]</a></li>

<li><a href="Joseph_Schillinger" title="Joseph Schillinger">Joseph Schillinger</a> (1895–1943) developed <a href="Schillinger_System" class="mw-redirect" title="Schillinger System">a system of composition</a> which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.<a href="#cite_note-FOOTNOTELivio2003193-97">[96]</a> See also <a href="Golden_ratio#Music" title="Golden ratio">Golden ratio § Music</a>.</li>

<li>In <a href="Software_development" title="Software development">software development</a>, Fibonacci numbers are often used by <a href="Agile_management" title="Agile management">agile</a> teams operating under the <a href="Scrum_(software_development)" title="Scrum (software development)">Scrum</a> framework to size their <a href="Product_backlog" title="Product backlog">product backlog</a> items.<a href="#cite_note-98">[97]</a></li>

<li>In the video game <a href="Genshin_Impact" title="Genshin Impact">Genshin Impact</a>'s soundtrack "<a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=DIclOmL1LGg">Gilded Runner</a>", the time signature follows the Fibonacci sequence.<a href="#cite_note-99">[98]</a></li></ul>

<ul><li><a href="The_Fibonacci_Association" title="The Fibonacci Association">The Fibonacci Association</a> – Organization for research on Fibonacci numbers</li>

<li><a href="Fibonacci_numbers_in_popular_culture" title="Fibonacci numbers in popular culture">Fibonacci numbers in popular culture</a></li>

<li><a href="Fibonacci_word" title="Fibonacci word">Fibonacci word</a> – Binary sequence from Fibonacci recurrence</li>

<li><a href="Random_Fibonacci_sequence" title="Random Fibonacci sequence">Random Fibonacci sequence</a> – Randomized mathematical sequence based upon the Fibonacci sequence</li>

<li><a href="Wythoff_array" title="Wythoff array">Wythoff array</a> – Infinite matrix of integers derived from the Fibonacci sequence</li>

<li><a href="International_Conference_on_Fibonacci_Numbers_and_their_Applications" title="International Conference on Fibonacci Numbers and their Applications">International Conference on Fibonacci Numbers and their Applications</a> – Mathematics conference</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Explanatory_footnotes">Explanatory footnotes</h3></div>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> "For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven <a href="Mora_(linguistics)" title="Mora (linguistics)">morae</a> [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <a href="#cite_note-13">[13]</a>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> Richard A. Brualdi, Introductory Combinatorics, Fifth edition, Pearson, 2005

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Peter Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994

</li>

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</style><cite id="CITEREFGoonatilake1998" class="citation cs2"><a href="Susantha_Goonatilake" title="Susantha Goonatilake">Goonatilake, Susantha</a> (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126">Toward a Global Science</a>, Indiana University Press, p. 126, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-253-33388-9</bdi></cite>

</li>

<li id="cite_note-HistoriaMathematica-4">^ <a href="#cite_ref-HistoriaMathematica_4-0">a</a> <a href="#cite_ref-HistoriaMathematica_4-1">b</a> <a href="#cite_ref-HistoriaMathematica_4-2">c</a> <cite id="CITEREFSingh1985" class="citation cs2">Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–244, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2885%2990021-7">10.1016/0315-0860(85)90021-7</a></cite>

</li>

<li id="cite_note-Donald_Knuth_2006_50-5">^ <a href="#cite_ref-Donald_Knuth_2006_50_5-0">a</a> <a href="#cite_ref-Donald_Knuth_2006_50_5-1">b</a> <cite id="CITEREFKnuth2006" class="citation cs2 cs1-prop-long-vol"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50">The Art of Computer Programming</a>, vol. 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-33570-8</bdi>, <q>it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when m = 7 are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)</q></cite>

</li>

<li id="cite_note-FOOTNOTESigler2002404–05-6"><a href="#cite_ref-FOOTNOTESigler2002404–05_6-0">^</a> <a href="#CITEREFSigler2002">Sigler 2002</a>, pp. 404–05.

</li>

<li id="cite_note-FOOTNOTELucas18913-7"><a href="#cite_ref-FOOTNOTELucas18913_7-0">^</a> <a href="#CITEREFLucas1891">Lucas 1891</a>, p. 3.

</li>

<li id="cite_note-FOOTNOTEBeckGeoghegan2010-8"><a href="#cite_ref-FOOTNOTEBeckGeoghegan2010_8-0">^</a> <a href="#CITEREFBeckGeoghegan2010">Beck & Geoghegan 2010</a>.

</li>

<li id="cite_note-FOOTNOTEBóna2011180-9"><a href="#cite_ref-FOOTNOTEBóna2011180_9-0">^</a> <a href="#CITEREFBóna2011">Bóna 2011</a>, p. 180.

</li>

<li id="cite_note-knuth-v1-10"><a href="#cite_ref-knuth-v1_10-0">^</a> <cite id="CITEREFKnuth1968" class="citation cs2"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1968), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100">The Art of Computer Programming</a>, vol. 1, Addison Wesley, p. 100, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-7758-754-8</bdi>, <q>Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed) ...</q></cite>

</li>

<li id="cite_note-FOOTNOTELivio2003197-11">^ <a href="#cite_ref-FOOTNOTELivio2003197_11-0">a</a> <a href="#cite_ref-FOOTNOTELivio2003197_11-1">b</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 197.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFAgrawala1969" class="citation cs2">Agrawala, VS (1969), Pāṇinikālīna Bhāratavarṣa (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan, <q>SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC</q></cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFVelankar1962" class="citation cs2">Velankar, HD (1962), 'Vṛttajātisamuccaya' of kavi Virahanka, Jodhpur: Rajasthan Oriental Research Institute, p. 101</cite>

</li>

<li id="cite_note-FOOTNOTELivio2003197–198-15"><a href="#cite_ref-FOOTNOTELivio2003197–198_15-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 197–198.

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFShah1991" class="citation cs2">Shah, Jayant (1991), <a rel="nofollow" class="external text" href="https://web.northeastern.edu/shah/papers/Pingala.pdf">A History of Piṅgala's Combinatorics</a> (PDF), <a href="Northeastern_University" title="Northeastern University">Northeastern University</a>, p. 41, retrieved 2019-01-04</cite>

</li>

<li id="cite_note-FOOTNOTESigler2002404–405-17"><a href="#cite_ref-FOOTNOTESigler2002404–405_17-0">^</a> <a href="#CITEREFSigler2002">Sigler 2002</a>, pp. 404–405.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html">"Fibonacci's Liber Abaci (Book of Calculation)"</a>, <a href="The_University_of_Utah" class="mw-redirect" title="The University of Utah">The University of Utah</a>, 13 December 2009, retrieved 2018-11-28</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFTassone1967" class="citation cs2">Tassone, Ann Dominic (April 1967), "A pair of rabbits and a mathematician", The Arithmetic Teacher, 14 (4): 285–288, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fat.14.4.0285">10.5951/at.14.4.0285</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41187298">41187298</a></cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFKnott" class="citation cs2">Knott, Ron, <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits">Fibonacci's Rabbits</a>, <a href="University_of_Surrey" title="University of Surrey">University of Surrey</a> Faculty of Engineering and Physical Sciences</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFGardner1996" class="citation cs2"><a href="Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (1996), Mathematical Circus, The Mathematical Association of America, p. 153, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-506-5</bdi>, <q>It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci</q></cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFbelcastro2018" class="citation book cs1"><a href="Sarah-Marie_Belcastro" class="mw-redirect" title="Sarah-Marie Belcastro">belcastro, sarah-marie</a> (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoqADwAAQBAJ">Discrete Mathematics with Ducks</a> (2nd ed.). CRC Press. p. 260. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-351-68369-2</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoqADwAAQBAJ&pg=PA260">Extract of page 260</a>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFBeutelspacherPetri1996" class="citation cs2">Beutelspacher, Albrecht; Petri, Bernhard (1996), "Fibonacci-Zahlen", Der Goldene Schnitt, Einblick in die Wissenschaft, Vieweg+Teubner Verlag, pp. 87–98, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-322-85165-9_6">10.1007/978-3-322-85165-9_6</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-8154-2511-4</bdi></cite>

</li>

<li id="cite_note-FOOTNOTEBall2003156-24"><a href="#cite_ref-FOOTNOTEBall2003156_24-0">^</a> <a href="#CITEREFBall2003">Ball 2003</a>, p. 156.

</li>

<li id="cite_note-FOOTNOTEBall2003155–156-25"><a href="#cite_ref-FOOTNOTEBall2003155–156_25-0">^</a> <a href="#CITEREFBall2003">Ball 2003</a>, pp. 155–156.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFSloane_"A002390"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A002390">"Sequence A002390 (Decimal expansion of natural logarithm of golden ratio)"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFSloane_"A097348"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A097348">"Sequence A097348 (Decimal expansion of arccsch(2)/log(10))"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFKepler1966" class="citation cs2">Kepler, Johannes (1966), A New Year Gift: On Hexagonal Snow, Oxford University Press, p. 92, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-858120-8</bdi></cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite class="citation cs2">Strena seu de Nive Sexangula, 1611</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFGessel1972" class="citation cs2">Gessel, Ira (October 1972), <a rel="nofollow" class="external text" href="https://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf">"Fibonacci is a Square"</a> (PDF), <a href="The_Fibonacci_Quarterly" class="mw-redirect" title="The Fibonacci Quarterly">The Fibonacci Quarterly</a>, 10 (4): 417–19, retrieved 2012-04-11</cite>

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://nrich.maths.org/2737">"The Golden Ratio, Fibonacci Numbers and Continued Fractions"</a>. nrich.maths.org. Retrieved 2024-03-22.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite id="CITEREFDijkstra1978" class="citation cs2"><a href="Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Dijkstra, Edsger W.</a> (1978), <a rel="nofollow" class="external text" href="https://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF">In honour of Fibonacci</a> (PDF)</cite>

</li>

<li id="cite_note-FOOTNOTELucas18914-33"><a href="#cite_ref-FOOTNOTELucas18914_33-0">^</a> <a href="#CITEREFLucas1891">Lucas 1891</a>, p. 4.

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFVorobievMartin2002" class="citation cs2">Vorobiev, Nikolaĭ Nikolaevich; Martin, Mircea (2002), "Chapter 1", Fibonacci Numbers, Birkhäuser, pp. 5–6, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-6135-8</bdi></cite>

</li>

<li id="cite_note-MathWorld-35">^ <a href="#cite_ref-MathWorld_35-0">a</a> <a href="#cite_ref-MathWorld_35-1">b</a> <a href="#cite_ref-MathWorld_35-2">c</a> <cite id="CITEREFWeisstein" class="citation web cs2"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FibonacciNumber.html">"Fibonacci Number"</a>, <a href="MathWorld" title="MathWorld">MathWorld</a></cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFGlaister1995" class="citation cs2">Glaister, P (1995), "Fibonacci power series", The Mathematical Gazette, 79 (486): 521–25, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3618079">10.2307/3618079</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3618079">3618079</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116536130">116536130</a></cite>

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="Edmund_Landau" title="Edmund Landau">Landau</a> (1899) quoted according <a href="#Borwein">Borwein</a>, Page 95, Exercise 3b.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFSloane_"A079586"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A079586">"Sequence A079586 (Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number)"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> <cite id="CITEREFAndré-Jeannin1989" class="citation cs2">André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", <a href="Comptes_Rendus_de_l'Acad%C3%A9mie_des_Sciences%2C_S%C3%A9rie_I" class="mw-redirect" title="Comptes Rendus de l'Académie des Sciences, Série I">Comptes Rendus de l'Académie des Sciences, Série I</a>, 308 (19): 539–41, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0999451">0999451</a></cite>

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFHonsberger1985" class="citation cs2">Honsberger, Ross (1985), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vl_0DwAAQBAJ&pg=PA135">"Millin's series"</a>, Mathematical Gems III, Dolciani Mathematical Expositions, vol. 9, American Mathematical Society, pp. 135–136, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781470457181</bdi></cite>

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFRibenboim2000" class="citation cs2"><a href="Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (2000), My Numbers, My Friends, Springer-Verlag</cite>

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> <cite id="CITEREFSu2000" class="citation cs2">Su, Francis E (2000), "Fibonacci GCD's, please", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091214092739/http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml">Mudd Math Fun Facts</a>, et al, HMC, archived from <a rel="nofollow" class="external text" href="http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml">the original</a> on 2009-12-14, retrieved 2007-02-23</cite>

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> <cite id="CITEREFWilliams1982" class="citation cs2">Williams, H. C. (1982), "A note on the Fibonacci quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{p-\varepsilon }/p}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{p-\varepsilon }/p}</annotation>

</semantics>

</math><img src="./ad9f956f8b55bc4bbd41da8ae361af6c076fbde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.93ex; height:3.009ex;" alt="{\displaystyle F_{p-\varepsilon }/p}" loading="lazy">", <a href="Canadian_Mathematical_Bulletin" title="Canadian Mathematical Bulletin">Canadian Mathematical Bulletin</a>, 25 (3): 366–70, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCMB-1982-053-0">10.4153/CMB-1982-053-0</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F137492">10338.dmlcz/137492</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0668957">0668957</a></cite>. Williams calls this property "well known".

</li>

<li id="cite_note-44"><a href="#cite_ref-44">^</a> Prime Numbers, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFSloane_"A005478"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A005478">"Sequence A005478 (Prime Fibonacci numbers)"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-46"><a href="#cite_ref-46">^</a> <cite id="CITEREFDiaconis2018" class="citation cs2"><a href="Persi_Diaconis" title="Persi Diaconis">Diaconis, Persi</a> (2018), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231118192225/https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf">"Probabilizing Fibonacci numbers"</a> (PDF), in <a href="Steve_Butler_(mathematician)" title="Steve Butler (mathematician)">Butler, Steve</a>; Cooper, Joshua; Hurlbert, Glenn (eds.), Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, pp. 1–12, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-15398-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3821829">3821829</a>, archived from <a rel="nofollow" class="external text" href="https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf">the original</a> (PDF) on 2023-11-18, retrieved 2022-11-23</cite>

</li>

<li id="cite_note-47"><a href="#cite_ref-47">^</a> <cite id="CITEREFHonsberger1985" class="citation cs2">Honsberger, Ross (1985), "Mathematical Gems III", AMS Dolciani Mathematical Expositions (9): 133, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-318-4</bdi></cite>

</li>

<li id="cite_note-48"><a href="#cite_ref-48">^</a> <cite id="CITEREFCohn1964" class="citation cs2">Cohn, J. H. E. (1964), "On square Fibonacci numbers", The Journal of the London Mathematical Society, 39: 537–540, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-39.1.537">10.1112/jlms/s1-39.1.537</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163867">0163867</a></cite>

</li>

<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFPethő2001" class="citation cs2">Pethő, Attila (2001), "Diophantine properties of linear recursive sequences II", Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 17: 81–96</cite>

</li>

<li id="cite_note-50"><a href="#cite_ref-50">^</a> <cite id="CITEREFBugeaudMignotteSiksek2006" class="citation cs2">Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0403046">math/0403046</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004math......3046B">2004math......3046B</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2006.163.969">10.4007/annals.2006.163.969</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10266596">10266596</a></cite>

</li>

<li id="cite_note-51"><a href="#cite_ref-51">^</a> <cite id="CITEREFLuo1989" class="citation cs2">Luo, Ming (1989), <a rel="nofollow" class="external text" href="https://www.fq.math.ca/Scanned/27-2/ming.pdf">"On triangular Fibonacci numbers"</a> (PDF), Fibonacci Quart., 27 (2): 98–108, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.1989.12429576">10.1080/00150517.1989.12429576</a></cite>

</li>

<li id="cite_note-Luca2000-52"><a href="#cite_ref-Luca2000_52-0">^</a> <cite id="CITEREFLuca2000" class="citation cs2">Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers", Rendiconti del Circolo Matematico di Palermo, 49 (2): 313–18, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02904236">10.1007/BF02904236</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1973-4409">1973-4409</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1765401">1765401</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121789033">121789033</a></cite>

</li>

<li id="cite_note-BGLLHT2011-53"><a href="#cite_ref-BGLLHT2011_53-0">^</a> <cite id="CITEREFBroughanGonzálezLewisLuca2011" class="citation cs2">Broughan, Kevin A.; González, Marcos J.; Lewis, Ryan H.; Luca, Florian; Mejía Huguet, V. Janitzio; Togbé, Alain (2011), <a rel="nofollow" class="external text" href="https://math.colgate.edu/~integers/vol11a.html">"There are no multiply-perfect Fibonacci numbers"</a>, Integers, 11a: A7, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2988067">2988067</a></cite>

</li>

<li id="cite_note-LucaMH2010-54"><a href="#cite_ref-LucaMH2010_54-0">^</a> <cite id="CITEREFLucaMejía_Huguet2010" class="citation cs2">Luca, Florian; Mejía Huguet, V. Janitzio (2010), <a rel="nofollow" class="external text" href="http://ami.ektf.hu/index.php?vol=37">"On Perfect numbers which are ratios of two Fibonacci numbers"</a>, Annales Mathematicae at Informaticae, 37: 107–24, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1787-6117">1787-6117</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2753031">2753031</a></cite>

</li>

<li id="cite_note-55"><a href="#cite_ref-55">^</a> <cite id="CITEREFKnott" class="citation cs2">Knott, Ron, <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html">The Fibonacci numbers</a>, UK: Surrey</cite>

</li>

<li id="cite_note-56"><a href="#cite_ref-56">^</a> <cite id="CITEREFSloane_"A235383"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A235383">"Sequence A235383 (Fibonacci numbers that are the product of other Fibonacci numbers)"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-57"><a href="#cite_ref-57">^</a> <cite id="CITEREFRibenboim1996" class="citation cs2"><a href="Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996), The New Book of Prime Number Records, New York: Springer, p. 64, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94457-9</bdi></cite>

</li>

<li id="cite_note-FOOTNOTELemmermeyer200073–74ex._2.25–28-58"><a href="#cite_ref-FOOTNOTELemmermeyer200073–74ex._2.25–28_58-0">^</a> <a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, pp. 73–74, ex. 2.25–28.

</li>

<li id="cite_note-FOOTNOTELemmermeyer200073–74ex._2.28-59"><a href="#cite_ref-FOOTNOTELemmermeyer200073–74ex._2.28_59-0">^</a> <a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, pp. 73–74, ex. 2.28.

</li>

<li id="cite_note-FOOTNOTELemmermeyer200073ex._2.27-60"><a href="#cite_ref-FOOTNOTELemmermeyer200073ex._2.27_60-0">^</a> <a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, p. 73, ex. 2.27.

</li>

<li id="cite_note-61"><a href="#cite_ref-61">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://mersennus.net/fibonacci/">Fibonacci and Lucas factorizations</a>, Mersennus</cite> collects all known factors of F(i) with i < 10000

</li>

<li id="cite_note-62"><a href="#cite_ref-62">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="http://fibonacci.redgolpe.com/">Factors of Fibonacci and Lucas numbers</a>, Red golpe</cite> collects all known factors of F(i) with 10000 < i < 50000

</li>

<li id="cite_note-63"><a href="#cite_ref-63">^</a> <cite id="CITEREFFreydBrown1993" class="citation cs2">Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–79, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2325076">10.2307/2325076</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2325076">2325076</a></cite>

</li>

<li id="cite_note-64"><a href="#cite_ref-64">^</a> <cite id="CITEREFSloane_"A001175"" class="citation web cs2"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A001175">"Sequence A001175 (Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n)"</a>, The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>, OEIS Foundation</cite>

</li>

<li id="cite_note-65"><a href="#cite_ref-65">^</a> <cite id="CITEREFLüWang2006" class="citation cs2">Lü, Kebo; Wang, Jun (2006), <a rel="nofollow" class="external text" href="https://utilitasmathematica.com/index.php/Index/article/view/410">"k-step Fibonacci sequence modulo m"</a>, Utilitas Mathematica, 71: 169–177, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2278830">2278830</a></cite>

</li>

<li id="cite_note-FOOTNOTELucas18917-66"><a href="#cite_ref-FOOTNOTELucas18917_66-0">^</a> <a href="#CITEREFLucas1891">Lucas 1891</a>, p. 7.

</li>

<li id="cite_note-67"><a href="#cite_ref-67">^</a> <cite id="CITEREFStanley2011" class="citation cs2">Stanley, Richard (2011), Enumerative Combinatorics I (2nd ed.), Cambridge Univ. Press, p. 121, Ex 1.35, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-60262-5</bdi></cite>

</li>

<li id="cite_note-68"><a href="#cite_ref-68">^</a> <cite id="CITEREFHarizanov1995" class="citation cs2"><a href="Valentina_Harizanov" title="Valentina Harizanov">Harizanov, Valentina</a> (1995), <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.rml/1204900767">"Review of Yuri V. Matiyasevich, Hibert's Tenth Problem"</a>, Modern Logic, 5 (3): 345–55</cite>

</li>

<li id="cite_note-69"><a href="#cite_ref-69">^</a> <cite id="CITEREFPagni2001" class="citation cs2">Pagni, David (September 2001), "Fibonacci Meets Pythagoras", Mathematics in School, 30 (4): 39–40, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30215477">30215477</a></cite>

</li>

<li id="cite_note-70"><a href="#cite_ref-70">^</a> <cite id="CITEREFStephenson2005" class="citation cs2">Stephenson, Kenneth (2005), <a href="Introduction_to_Circle_Packing" title="Introduction to Circle Packing">Introduction to Circle Packing: The Theory of Discrete Analytic Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-82356-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2131318">2131318</a></cite>; see especially Lemma 8.2 (Ring Lemma), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=38PxEmKKhysC&pg=PA73">pp. 73–74</a>, and Appendix B, The Ring Lemma, pp. 318–321.

</li>

<li id="cite_note-71"><a href="#cite_ref-71">^</a> <cite id="CITEREFKnuth1997" class="citation cs2 cs1-prop-long-vol"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E</a> (1997), The Art of Computer Programming, vol. 1: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-89683-1</bdi></cite>

</li>

<li id="cite_note-72"><a href="#cite_ref-72">^</a> <cite id="CITEREFAdelson-VelskyLandis1962" class="citation cs2 cs1-prop-foreign-lang-source">Adelson-Velsky, Georgy; Landis, Evgenii (1962), "An algorithm for the organization of information", <a href="Proceedings_of_the_USSR_Academy_of_Sciences" title="Proceedings of the USSR Academy of Sciences">Proceedings of the USSR Academy of Sciences</a> (in Russian), 146: 263–266</cite> <a rel="nofollow" class="external text" href="https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf">English translation</a> by Myron J. Ricci in Soviet Mathematics - Doklady, 3:1259–1263, 1962.

</li>

<li id="cite_note-73"><a href="#cite_ref-73">^</a> <cite id="CITEREFAvrielWilde1966" class="citation cs2">Avriel, M; Wilde, DJ (1966), "Optimality of the Symmetric Fibonacci Search Technique", Fibonacci Quarterly (3): 265–69, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.1966.12431364">10.1080/00150517.1966.12431364</a></cite>

</li>

<li id="cite_note-74"><a href="#cite_ref-74">^</a> <cite class="citation cs2">Amiga ROM Kernel Reference Manual, Addison–Wesley, 1991</cite>

</li>

<li id="cite_note-75"><a href="#cite_ref-75">^</a> <cite class="citation cs2">"IFF", <a rel="nofollow" class="external text" href="https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression">Multimedia Wiki</a></cite>

</li>

<li id="cite_note-76"><a href="#cite_ref-76">^</a> <cite id="CITEREFDean_Leffingwell2021" class="citation cs2">Dean Leffingwell (2021-07-01), <a rel="nofollow" class="external text" href="https://www.scaledagileframework.com/story/">Story</a>, Scaled Agile Framework, retrieved 2022-08-15</cite>

</li>

<li id="cite_note-77"><a href="#cite_ref-77">^</a> <cite id="CITEREFDouadyCouder1996" class="citation cs2">Douady, S; Couder, Y (1996), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf">"Phyllotaxis as a Dynamical Self Organizing Process"</a> (PDF), Journal of Theoretical Biology, 178 (3): 255–74, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjtbi.1996.0026">10.1006/jtbi.1996.0026</a>, archived from <a rel="nofollow" class="external text" href="http://www.math.ntnu.no/~jarlet/Douady96.pdf">the original</a> (PDF) on 2006-05-26</cite>

</li>

<li id="cite_note-78"><a href="#cite_ref-78">^</a> <cite id="CITEREFJonesWilson2006" class="citation cs2">Jones, Judy; Wilson, William (2006), "Science", An Incomplete Education, Ballantine Books, p. 544, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7394-7582-9</bdi></cite>

</li>

<li id="cite_note-79"><a href="#cite_ref-79">^</a> <cite id="CITEREFBrousseau1969" class="citation cs2">Brousseau, A (1969), "Fibonacci Statistics in Conifers", <a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a> (7): 525–32</cite>

</li>

<li id="cite_note-80"><a href="#cite_ref-80">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.cs4fn.org/maths/bee-davinci.php">"Marks for the da Vinci Code: B–"</a>, Maths, Computer Science For Fun: CS4FN</cite>

</li>

<li id="cite_note-81"><a href="#cite_ref-81">^</a> <cite id="CITEREFScottMarketos2014" class="citation cs2">Scott, T.C.; Marketos, P. (March 2014), <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf">On the Origin of the Fibonacci Sequence</a> (PDF), <a href="MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>, University of St Andrews</cite>

</li>

<li id="cite_note-FOOTNOTELivio2003110-82"><a href="#cite_ref-FOOTNOTELivio2003110_82-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 110.

</li>

<li id="cite_note-FOOTNOTELivio2003112–13-83"><a href="#cite_ref-FOOTNOTELivio2003112–13_83-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, pp. 112–13.

</li>

<li id="cite_note-84"><a href="#cite_ref-84">^</a> <cite id="CITEREFVarenne2010" class="citation cs2 cs1-prop-foreign-lang-source">Varenne, Franck (2010), <a rel="nofollow" class="external text" href="https://www.numilog.com/LIVRES/ISBN/9782705670894.Livre">Formaliser le vivant - Lois, Théories, Modèles</a> (in French), Hermann, p. 28, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9782705678128</bdi>, retrieved 2022-10-30, <q>En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].</q></cite>

</li>

<li id="cite_note-85"><a href="#cite_ref-85">^</a> <cite id="CITEREFPrusinkiewiczHanan1989" class="citation cs2">Prusinkiewicz, Przemyslaw; Hanan, James (1989), Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics), <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-97092-9</bdi></cite>

</li>

<li id="cite_note-86"><a href="#cite_ref-86">^</a> <cite id="CITEREFVogel1979" class="citation cs2">Vogel, Helmut (1979), "A better way to construct the sunflower head", Mathematical Biosciences, 44 (3–4): 179–89, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0025-5564%2879%2990080-4">10.1016/0025-5564(79)90080-4</a></cite>

</li>

<li id="cite_note-FOOTNOTELivio2003112-87"><a href="#cite_ref-FOOTNOTELivio2003112_87-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 112.

</li>

<li id="cite_note-88"><a href="#cite_ref-88">^</a> <cite id="CITEREFPrusinkiewiczLindenmayer1990" class="citation cs2"><a href="Przemyslaw_Prusinkiewicz" class="mw-redirect" title="Przemyslaw Prusinkiewicz">Prusinkiewicz, Przemyslaw</a>; <a href="Aristid_Lindenmayer" title="Aristid Lindenmayer">Lindenmayer, Aristid</a> (1990), <a rel="nofollow" class="external text" href="https://algorithmicbotany.org/papers/#webdocs">"4"</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/algorithmicbeaut0000prus/page/101">The Algorithmic Beauty of Plants</a>, Springer-Verlag, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/algorithmicbeaut0000prus/page/101">101–107</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-97297-8</bdi></cite>

</li>

<li id="cite_note-89"><a href="#cite_ref-89">^</a> <cite id="CITEREFBasin1963" class="citation cs2">Basin, S. L. (1963), <a rel="nofollow" class="external text" href="https://www.fq.math.ca/Scanned/1-1/basin.pdf">"The Fibonacci sequence as it appears in nature"</a> (PDF), The Fibonacci Quarterly, 1 (1): 53–56, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.1963.12431602">10.1080/00150517.1963.12431602</a></cite>

</li>

<li id="cite_note-90"><a href="#cite_ref-90">^</a> Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.

</li>

<li id="cite_note-xcs-91">^ <a href="#cite_ref-xcs_91-0">a</a> <a href="#cite_ref-xcs_91-1">b</a> <cite id="CITEREFHutchison2004" class="citation cs2">Hutchison, Luke (September 2004), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf">"Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships"</a> (PDF), Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04), archived from <a rel="nofollow" class="external text" href="http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf">the original</a> (PDF) on 2020-09-25, retrieved 2016-09-03</cite>

</li>

<li id="cite_note-FOOTNOTELivio200398–99-92"><a href="#cite_ref-FOOTNOTELivio200398–99_92-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, pp. 98–99.

</li>

<li id="cite_note-93"><a href="#cite_ref-93">^</a> <cite class="citation cs2">"Zeckendorf representation", <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Zeckendorf_representation">Encyclopedia of Math</a></cite>

</li>

<li id="cite_note-94"><a href="#cite_ref-94">^</a> <cite id="CITEREFPatranabisDana1985" class="citation cs2 cs1-prop-long-vol">Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers", <a href="IEEE_Transactions_on_Instrumentation_and_Measurement" title="IEEE Transactions on Instrumentation and Measurement">IEEE Transactions on Instrumentation and Measurement</a>, IM-34 (4): 650–653, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985ITIM...34..650P">1985ITIM...34..650P</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftim.1985.4315428">10.1109/tim.1985.4315428</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35413237">35413237</a></cite>

</li>

<li id="cite_note-Brasch_et_al._2012-95"><a href="#cite_ref-Brasch_et_al._2012_95-0">^</a> <cite id="CITEREFBraschByströmLystad2012" class="citation cs2">Brasch, T. von; Byström, J.; Lystad, L.P. (2012), <a rel="nofollow" class="external text" href="https://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-24073">"Optimal Control and the Fibonacci Sequence"</a>, Journal of Optimization Theory and Applications, 154 (3): 857–78, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10957-012-0061-2">10.1007/s10957-012-0061-2</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11250%2F180781">11250/180781</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8550726">8550726</a></cite>

</li>

<li id="cite_note-FOOTNOTELivio2003176-96"><a href="#cite_ref-FOOTNOTELivio2003176_96-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 176.

</li>

<li id="cite_note-FOOTNOTELivio2003193-97"><a href="#cite_ref-FOOTNOTELivio2003193_97-0">^</a> <a href="#CITEREFLivio2003">Livio 2003</a>, p. 193.

</li>

<li id="cite_note-98"><a href="#cite_ref-98">^</a> <cite id="CITEREFKathuria" class="citation web cs1">Kathuria, Madhur. <a rel="nofollow" class="external text" href="https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum">"A Guide to Using the Fibonacci Sequence in Scrum"</a>. Scrum Alliance. Retrieved 8 August 2025.</cite>

</li>

<li id="cite_note-99"><a href="#cite_ref-99">^</a> <cite id="CITEREFGenshin_Impact2022" class="citation audio-visual cs1">Genshin Impact (2022-10-22). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ps8oa3CRNfk">"Travelers' Reverie" — Behind the Scenes of the Music of Sumeru | Genshin Impact</a>. Retrieved 2025-08-11 – via YouTube.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3></div>

<ul><li><cite id="CITEREFBall2003" class="citation cs2">Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited", Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Princeton, NJ: <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-11321-0</bdi></cite>.</li>

<li><cite id="CITEREFBeckGeoghegan2010" class="citation cs2">Beck, Matthias; Geoghegan, Ross (2010), The Art of Proof: Basic Training for Deeper Mathematics, New York: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-7022-0</bdi></cite>.</li>

<li><cite id="CITEREFBóna2011" class="citation cs2"><a href="Mikl%C3%B3s_B%C3%B3na" title="Miklós Bóna">Bóna, Miklós</a> (2011), A Walk Through Combinatorics (3rd ed.), New Jersey: World Scientific, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-4335-23-2</bdi></cite>.</li>

<li><cite id="CITEREFBorweinBorwein1998" class="citation cs2"><a href="Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan M.</a>; <a href="Peter_Borwein" title="Peter Borwein">Borwein, Peter B.</a> (July 1998), <a rel="nofollow" class="external text" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-047131515X.html">Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</a>, Wiley, pp. 91–101, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-31515-5</bdi></cite></li>

<li><cite id="CITEREFLemmermeyer2000" class="citation cs2">Lemmermeyer, Franz (2000), Reciprocity Laws: From Euler to Eisenstein, Springer Monographs in Mathematics, New York: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-66957-9</bdi></cite>.</li>

<li><cite id="CITEREFLivio2003" class="citation cs2"><a href="Mario_Livio" title="Mario Livio">Livio, Mario</a> (2003) [2002], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bUARfgWRH14C">The Golden Ratio: The Story of Phi, the World's Most Astonishing Number</a> (First trade paperback ed.), New York City: <a href="Random_House" title="Random House">Broadway Books</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7679-0816-3</bdi></cite></li>

<li><cite id="CITEREFLucas1891" class="citation cs2 cs1-prop-foreign-lang-source">Lucas, Édouard (1891), <a rel="nofollow" class="external text" href="https://archive.org/details/thoriedesnombr01lucauoft">Théorie des nombres</a> (in French), vol. 1, Paris: Gauthier-Villars</cite>.</li>

<li><cite id="CITEREFSigler2002" class="citation cs2">Sigler, L. E. (2002), Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95419-6</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikiquote has quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Fibonacci_sequence" class="extiw external" title="q:Special:Search/Fibonacci sequence">Fibonacci sequence</a>.</div></div>

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<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Fibonacci_number_program" class="extiw external" title="wikibooks:Fibonacci number program">Fibonacci number program</a></div></div>

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<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=hbUQlrLDAgw">Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram</a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> - animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy</li>

<li><a rel="nofollow" class="external text" href="https://www.mathpages.com/home/kmath078/kmath078.htm">Periods of Fibonacci Sequences Mod m</a> at MathPages</li>

<li><a rel="nofollow" class="external text" href="http://www.physorg.com/news97227410.html">Scientists find clues to the formation of Fibonacci spirals in nature</a></li>

<li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b008ct2j">Fibonacci Sequence</a> on <a href="In_Our_Time_(radio_series)" title="In Our Time (radio series)">In Our Time</a> at the <a href="BBC" title="BBC">BBC</a></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Fibonacci_numbers">"Fibonacci numbers"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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</style><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"><a href="Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Power_of_two" title="Power of two">Power of 2</a></li>

<li><a href="Power_of_three" title="Power of three">Power of 3</a></li>

<li><a href="Power_of_10" title="Power of 10">Power of 10</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Fourth_power" title="Fourth power">Fourth power</a></li>

<li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a></li>

<li><a href="Seventh_power" title="Seventh power">Seventh power</a></li>

<li><a href="Eighth_power" title="Eighth power">Eighth power</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Prime_power" title="Prime power">Prime power</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form a × 2b ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cullen_number" title="Cullen number">Cullen</a></li>

<li><a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li>

<li><a href="Fermat_number" title="Fermat number">Fermat</a></li>

<li><a href="Mersenne_prime" title="Mersenne prime">Mersenne</a></li>

<li><a href="Proth_number" class="mw-redirect" title="Proth number">Proth</a></li>

<li><a href="Thabit_number" title="Thabit number">Thabit</a></li>

<li><a href="Woodall_number" title="Woodall number">Woodall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hilbert_number" title="Hilbert number">Hilbert</a></li>

<li><a href="Idoneal_number" title="Idoneal number">Idoneal</a></li>

<li><a href="Leyland_number" title="Leyland number">Leyland</a></li>

<li><a href="Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li>

<li><a href="Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li>

<li><a href="Leonardo_number" title="Leonardo number">Leonardo</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas</a></li>

<li><a href="Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li>

<li><a href="Padovan_sequence" title="Padovan sequence">Padovan</a></li>

<li><a href="Perrin_number" title="Perrin number">Perrin</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_number" title="Amenable number">Amenable</a></li>

<li><a href="Congruent_number" title="Congruent number">Congruent</a></li>

<li><a href="Kn%C3%B6del_number" title="Knödel number">Knödel</a></li>

<li><a href="Riesel_number" title="Riesel number">Riesel</a></li>

<li><a href="Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li>

<li><a href="Polite_number" title="Polite number">Polite</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li>

<li><a href="Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Triangular_number" title="Triangular number">Triangular</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Square_triangular_number" title="Square triangular number">Square triangular</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Catalan_number" title="Catalan number">Catalan</a></li>

<li><a href="Dedekind_number" title="Dedekind number">Dedekind</a></li>

<li><a href="Delannoy_number" title="Delannoy number">Delannoy</a></li>

<li><a href="Euler_number" class="mw-redirect" title="Euler number">Euler</a></li>

<li><a href="Eulerian_number" title="Eulerian number">Eulerian</a></li>

<li><a href="Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li>

<li><a href="Lazy_caterer's_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li>

<li><a href="Motzkin_number" title="Motzkin number">Motzkin</a></li>

<li><a href="Narayana_number" title="Narayana number">Narayana</a></li>

<li><a href="Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li>

<li><a href="Schr%C3%B6der_number" title="Schröder number">Schröder</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li>

<li><a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li>

<li><a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li>

<li><a href="Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li>

<li><a href="Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li>

<li><a href="Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li>

<li><a href="Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carmichael_number" title="Carmichael number">Carmichael number</a></li>

<li><a href="Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li>

<li><a href="Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li>

<li><a href="Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li>

<li><a href="Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li>

<li><a href="Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li>

<li><a href="Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li>

<li><a href="Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li>

<li><a href="Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li>

<li><a href="Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li>

<li><a href="Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_prime" title="Almost prime">Almost prime</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li>

<li><a href="Highly_totient_number" title="Highly totient number">Highly totient</a></li>

<li><a href="Noncototient" title="Noncototient">Noncototient</a></li>

<li><a href="Nontotient" title="Nontotient">Nontotient</a></li>

<li><a href="Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li>

<li><a href="Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li>

<li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid_number" title="Euclid number">Euclid</a></li>

<li><a href="Fortunate_number" title="Fortunate number">Fortunate</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li>

<li><a href="Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Giuga_number" title="Giuga number">Giuga</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="St%C3%B8rmer_number" title="Størmer number">Størmer</a></li>

<li><a href="Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Persistence_of_a_number" title="Persistence of a number">Persistence</a>

<ul><li><a href="Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li>

<li><a href="Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digit_sum" title="Digit sum">Digit sum</a></li>

<li><a href="Digital_root" title="Digital root">Digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meertens_number" title="Meertens number">Meertens</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dudeney_number" title="Dudeney number">Dudeney</a></li>

<li><a href="Factorion" title="Factorion">Factorion</a></li>

<li><a href="Kaprekar_number" title="Kaprekar number">Kaprekar</a></li>

<li><a href="Kaprekar's_routine" title="Kaprekar's routine">Kaprekar's constant</a></li>

<li><a href="Keith_number" title="Keith number">Keith</a></li>

<li><a href="Lychrel_number" title="Lychrel number">Lychrel</a></li>

<li><a href="Narcissistic_number" title="Narcissistic number">Narcissistic</a></li>

<li><a href="Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li>

<li><a href="Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a>

<ul><li><a href="Happy_number" title="Happy number">Happy</a></li></ul></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automorphic_number" title="Automorphic number">Automorphic</a>

<ul><li><a href="Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Palindromic_number" title="Palindromic number">Palindromic</a></li>

<li><a href="Pandigital_number" title="Pandigital number">Pandigital</a></li>

<li><a href="Repdigit" title="Repdigit">Repdigit</a></li>

<li><a href="Repunit" title="Repunit">Repunit</a></li>

<li><a href="Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li>

<li><a href="Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li>

<li><a href="Undulating_number" title="Undulating number">Undulating</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cyclic_number" title="Cyclic number">Cyclic</a></li>

<li><a href="Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li>

<li><a href="Parasitic_number" title="Parasitic number">Parasitic</a></li>

<li><a href="Primeval_number" title="Primeval number">Primeval</a></li>

<li><a href="Transposable_integer" title="Transposable integer">Transposable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li>

<li><a href="Vampire_number" title="Vampire number">Vampire</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Friedman_number" title="Friedman number">Friedman</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Odious_number" title="Odious number">Odious</a></li>

<li><a href="Pernicious_number" title="Pernicious number">Pernicious</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lucky_number" title="Lucky number">Lucky</a></li>

<li><a href="Generation_of_primes" title="Generation of primes">Prime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pancake_sorting" title="Pancake sorting">Pancake number</a></li>

<li><a href="Sorting_number" title="Sorting number">Sorting number</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aronson's_sequence" title="Aronson's sequence">Aronson's sequence</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div>

<ul><li> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Metallic_means18" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Metallic_means18" style="font-size:114%;margin:0 4em"><a href="Metallic_mean" title="Metallic mean">Metallic means</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot number</a></li>

<ul><li><a href="Golden_angle" title="Golden angle">Angle</a></li>

<li><a href="Kepler_triangle" title="Kepler triangle">Kepler triangle</a></li>

<li><a href="Golden_rectangle" title="Golden rectangle">Rectangle</a></li>

<li><a href="Golden_rhombus" title="Golden rhombus">Rhombus</a></li>

<li><a href="Golden-section_search" title="Golden-section search">Section search</a></li>

<li><a href="Golden_spiral" title="Golden spiral">Spiral</a></li>

<li><a href="Golden_triangle_(mathematics)" title="Golden triangle (mathematics)">Triangle</a></li>

<li><a href="Supergolden_ratio" title="Supergolden ratio">Supergolden ratio</a></li></ul></li></ul>

<ul><li><a href="Silver_ratio" title="Silver ratio">Silver</a>

<ul><li><a href="Pell_number" title="Pell number">Pell number</a></li>

<li><a href="Supersilver_ratio" title="Supersilver ratio">Supersilver ratio</a></li></ul></li>

<li>Bronze</li>

<li>Copper</li>

<li>Nickel</li>

</div></td><td class="noviewer navbox-image" rowspan="1" style="width:1px;padding:0 0 0 2px"><div></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Sequences_and_series332" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Sequences_and_series332" style="font-size:114%;margin:0 4em"><a href="Sequence" title="Sequence">Sequences</a> and <a href="Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li>

<li><a href="Geometric_progression" title="Geometric progression">Geometric progression</a></li>

<li><a href="Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li>

<li><a href="Square_number" title="Square number">Square number</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Power_of_two" title="Power of two">Powers of two</a></li>

<li><a href="Power_of_three" title="Power of three">Powers of three</a></li>

<li><a href="Power_of_10" title="Power of 10">Powers of 10</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced (<a href="List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complete_sequence" title="Complete sequence">Complete sequence</a></li>

<li><a href="Figurate_number" title="Figurate number">Figurate number</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas number</a></li>

<li><a href="Pell_number" title="Pell number">Pell number</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li>

<li><a href="Polygonal_number" title="Polygonal number">Polygonal number</a></li>

<li><a href="Triangular_number" title="Triangular number">Triangular number</a>

<ul><li><a href="Triangular_array" title="Triangular array">array</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="Fibonacci_sequence" title="Fibonacci sequence"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Monotonic_function" title="Monotonic function">Monotonic function</a></li>

<li><a href="Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Convergent_series" title="Convergent series">Convergent</a></li>

<li><a href="Divergent_series" title="Divergent series">Divergent</a></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_convergence" title="Absolute convergence">Absolute</a></li>

<li><a href="Conditional_convergence" title="Conditional convergence">Conditional</a></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Riemann_zeta_function" title="Riemann zeta function">1 + 1/2s + 1/3s + ... (Riemann zeta function)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Grandi's_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li>

<li><a href="Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li>

<li><a href="1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li>

<li><a href="Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Taylor_series" title="Taylor series">Taylor series</a></li>

<li><a href="Power_series" title="Power series">Power series</a></li>

<li><a href="Formal_power_series" title="Formal power series">Formal power series</a></li>

<li><a href="Laurent_series" title="Laurent series">Laurent series</a></li>

<li><a href="Puiseux_series" title="Puiseux series">Puiseux series</a></li>

<li><a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li>

<li><a href="Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier series</a></li>

<li><a href="Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li>

<li><a href="Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li>

<li><a href="Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li>

<li><a href="Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li>

<li><a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li>

<li><a href="Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Fibonacci33" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Fibonacci33" style="font-size:114%;margin:0 4em"><a href="Fibonacci" title="Fibonacci">Fibonacci</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Books</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Liber_Abaci" title="Liber Abaci">Liber Abaci</a> (1202)</li>

<li><a href="The_Book_of_Squares" title="The Book of Squares">The Book of Squares</a> (1225)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Greedy_algorithm_for_Egyptian_fractions" title="Greedy algorithm for Egyptian fractions">Greedy algorithm for Egyptian fractions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fibonacci_numbers_in_popular_culture" title="Fibonacci numbers in popular culture">Fibonacci numbers in popular culture</a></li>

<li><a href="List_of_things_named_after_Fibonacci" title="List of things named after Fibonacci">List of things named after Fibonacci</a></li>

<li><a href="Generalizations_of_Fibonacci_numbers" title="Generalizations of Fibonacci numbers">Generalizations of Fibonacci numbers</a></li>

<li><a href="The_Fibonacci_Association" title="The Fibonacci Association">The Fibonacci Association</a>

<ul><li><a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></li></ul></li></ul>

</div></td></tr></tbody></table></div>

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