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<title>Squared triangular number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Squared triangular number</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For triangular numbers that are themselves square, see <a href="Square_triangular_number" title="Square triangular number">square triangular number</a>.</div>

<p>In <a href="Number_theory" title="Number theory">number theory</a>, the sum of the first <span class="texhtml mvar" style="font-style:italic;">n</span> <a href="Cube_(algebra)" title="Cube (algebra)">cubes</a> is the <a href="Square_number" title="Square number">square</a> of the <span class="texhtml mvar" style="font-style:italic;">n</span>th <a href="Triangular_number" title="Triangular number">triangular number</a>. That is,

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

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</math></span><img src="./e4fc1d31596dd03e96bc0430bab89165792f7aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.76ex; height:3.343ex;" alt="{\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}.}" loading="lazy"></span></dd></dl>

<p>The same equation may be written more compactly using the mathematical notation for <a href="Summation" title="Summation">summation</a>:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}</annotation>

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</math></span><img src="./4ba2e42d8eb4f44395e0516aed3da998ed86bd3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.442ex; height:8.009ex;" alt="{\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}" loading="lazy"></span></dd></dl>

<p>This <a href="Identity_(mathematics)" title="Identity (mathematics)">identity</a> is sometimes called <b>Nicomachus's theorem</b>, after <a href="Nicomachus" title="Nicomachus">Nicomachus of Gerasa</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 60</span> – <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 120 CE</span>).

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

<p>Nicomachus, at the end of Chapter 20 of his <i>Introduction to Arithmetic</i>, pointed out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the first <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> cubes equals the sum of the first <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n(n+1)}{2}}}">

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

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</math></span><img src="./e1dc30e5c6b7a621f43c768266851d4cf953124c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n(n+1)}{2}}}" loading="lazy"></span> odd numbers, that is, the odd numbers from 1 to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(n+1)-1}">

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

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</math></span><img src="./75b6fa30b80ef27b195c2d77b84eaae95b6eac54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.604ex; height:2.843ex;" alt="{\displaystyle n(n+1)-1}" loading="lazy"></span>. The average of these numbers is obviously <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n(n+1)}{2}}}">

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

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</math></span><img src="./e1dc30e5c6b7a621f43c768266851d4cf953124c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n(n+1)}{2}}}" loading="lazy"></span>, and there are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n(n+1)}{2}}}">

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

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</math></span><img src="./e1dc30e5c6b7a621f43c768266851d4cf953124c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n(n+1)}{2}}}" loading="lazy"></span> of them, so their sum is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {n(n+1)}{2}}\right)^{2}}">

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

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</math></span><img src="./4ba64edd371445c948bed6ec0cfa1fde9b491c0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.018ex; height:5.176ex;" alt="{\displaystyle \left({\tfrac {n(n+1)}{2}}\right)^{2}}" loading="lazy"></span>.

</p><p>Many early mathematicians have studied and provided proofs of Nicomachus's theorem. <a href="#CITEREFStroeker1995">Stroeker (1995)</a> claims that "every student of number theory surely must have marveled at this miraculous fact".<sup id="cite_ref-FOOTNOTEStroeker1995_1-0" class="reference"><a href="#cite_note-FOOTNOTEStroeker1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFPengelley2002">Pengelley (2002)</a> finds references to the identity not only in the works of <a href="Nicomachus" title="Nicomachus">Nicomachus</a> in what is now <a href="Jordan" title="Jordan">Jordan</a> in the 1st century CE, but also in those of <a href="Aryabhata" title="Aryabhata">Aryabhata</a> in <a href="India" title="India">India</a> in the 5th century, and in those of <a href="Al-Karaji" title="Al-Karaji">Al-Karaji</a> <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1000</span> in <a href="Persia" class="mw-redirect" title="Persia">Persia</a>.<sup id="cite_ref-FOOTNOTEPengelley2002_2-0" class="reference"><a href="#cite_note-FOOTNOTEPengelley2002-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFBressoud2004">Bressoud (2004)</a> mentions several additional early mathematical works on this formula, by <a href="Al-Qabisi" title="Al-Qabisi">Al-Qabisi</a> (10th century Arabia), <a href="Gersonides" title="Gersonides">Gersonides</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1300</span>, France), and <a href="Nilakantha_Somayaji" title="Nilakantha Somayaji">Nilakantha Somayaji</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1500</span>, India); he reproduces Nilakantha's visual proof.<sup id="cite_ref-FOOTNOTEBressoud2004_3-0" class="reference"><a href="#cite_note-FOOTNOTEBressoud2004-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Numeric_values;_geometric_and_probabilistic_interpretation">Numeric values; geometric and probabilistic interpretation</h2></div>

<p>The sequence of squared triangular numbers is

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</style><div class="block-indent" style="padding-left: 1.6em;"><span class="texhtml"><a href="0_(number)" class="mw-redirect" title="0 (number)">0</a>, <a href="1_(number)" class="mw-redirect" title="1 (number)">1</a>, <a href="9_(number)" class="mw-redirect" title="9 (number)">9</a>, <a href="36_(number)" title="36 (number)">36</a>, <a href="100_(number)" class="mw-redirect" title="100 (number)">100</a>, 225,</span> <span class="texhtml">441, 784, 1296, 2025, 3025, 4356, 6084, 8281,</span> ... (sequence <span class="nowrap external"><a href="https://oeis.org/A000537" class="extiw external" title="oeis:A000537">A000537</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</div>

<p>These numbers can be viewed as <a href="Figurate_number" title="Figurate number">figurate numbers</a>, a four-dimensional hyperpyramidal generalization of the <a href="Triangular_number" title="Triangular number">triangular numbers</a> and <a href="Square_pyramidal_number" title="Square pyramidal number">square pyramidal numbers</a>.

</p><p>As <a href="#CITEREFStein1971">Stein (1971)</a> observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> <a href="Square_lattice" title="Square lattice">grid</a>. For instance, the points of a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\times 4}">

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</math></span><img src="./89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 4}" loading="lazy"></span> grid (or a square made up of three smaller squares on a side) can form 36 different rectangles. The number of squares in a square grid is similarly counted by the square pyramidal numbers.<sup id="cite_ref-FOOTNOTEStein1971_4-0" class="reference"><a href="#cite_note-FOOTNOTEStein1971-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>The identity also admits a natural probabilistic interpretation as follows. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y,Z,W}">

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<annotation encoding="application/x-tex">{\displaystyle X,Y,Z,W}</annotation>

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</math></span><img src="./26393dbf27b7da293fe1906c3dddf3ed4c5731c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.971ex; height:2.509ex;" alt="{\displaystyle X,Y,Z,W}" loading="lazy"></span> be four integer numbers independently and uniformly chosen at random between 1 and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>. Then, the probability that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is the largest of the four numbers equals the probability that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> is at least as large as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is at least as large as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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

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</math></span><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span>. That is, <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[\max(X,Y,Z)\leq W]=\Pr[X\leq Y\wedge Z\leq W].}">

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</math></span></span> For any particular value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>, the combinations of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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

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</math></span><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> that make <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> largest form a cube <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq X,Y,Z\leq n}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq X,Y,Z\leq n}</annotation>

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</math></span><img src="./afe6120fd117a574522a5e1a8adecef3fd120fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.256ex; height:2.509ex;" alt="{\displaystyle 1\leq X,Y,Z\leq n}" loading="lazy"></span> so (adding the size of this cube over all choices of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span>}) the number of combinations of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y,Z,W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,Y,Z,W}</annotation>

</semantics>

</math></span><img src="./26393dbf27b7da293fe1906c3dddf3ed4c5731c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.971ex; height:2.509ex;" alt="{\displaystyle X,Y,Z,W}" loading="lazy"></span> for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is largest is a sum of cubes, the left hand side of the Nichomachus identity. The sets of pairs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation>

</semantics>

</math></span><img src="./41f29b9537685f499713112d6802e811cbf51bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\leq Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\leq Y}</annotation>

</semantics>

</math></span><img src="./a8ce0a0bde4c0fecbb48124e9cd3bd9669ac6719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle X\leq Y}" loading="lazy"></span> and of pairs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Z,W)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Z,W)}</annotation>

</semantics>

</math></span><img src="./745068b2a224e0b8b3ee2815aefe028e3b4eca0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.959ex; height:2.843ex;" alt="{\displaystyle (Z,W)}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\leq W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\leq W}</annotation>

</semantics>

</math></span><img src="./278f8d2c156d2b0ec01ceda28731aed20cf5de0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.214ex; height:2.343ex;" alt="{\displaystyle Z\leq W}" loading="lazy"></span> form isosceles right triangles, and the set counted by the right hand side of the equation of probabilities is the <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of these two triangles, so its size is the square of a triangular number on the right hand side of the Nichomachus identity. The probabilities themselves are respectively the left and right sides of the Nichomachus identity, normalized to make probabilities by dividing both sides <span class="nowrap">by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{4}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6c1d9f4962c4e7493552104d007b61e8639e5d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{4}}" loading="lazy"></span>.</span>

</p>

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

<p><a href="Charles_Wheatstone" title="Charles Wheatstone">Charles Wheatstone</a> (<a href="#CITEREFWheatstone1854">1854</a>) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. He begins by giving the identity

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mrow>

<mtext> consecutive odd numbers</mtext>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n^{3}=\underbrace {\left(n^{2}-n+1\right)+\left(n^{2}-n+1+2\right)+\left(n^{2}-n+1+4\right)+\cdots +\left(n^{2}+n-1\right)} _{n{\text{ consecutive odd numbers}}}.}</annotation>

</semantics>

</math></span></span>

That identity is related to <a href="Triangular_numbers" class="mw-redirect" title="Triangular numbers">triangular numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy"></span> in the following way:

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n^{3}=\sum _{k=T_{n-1}+1}^{T_{n}}(2k-1),}</annotation>

</semantics>

</math></span></span>

and thus the summands forming <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3e9d1a52e455a7a5272a345b2697e35f1579b681.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{3}}" loading="lazy"></span> start off just after those forming all previous values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c815cf8948a410cd19c41a251afa2724394882a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 1^{3}}" loading="lazy"></span> up to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./732b04063004f6e14e1e1dfe290c107e2810a507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.261ex; height:3.176ex;" alt="{\displaystyle (n-1)^{3}}" loading="lazy"></span>. Applying this property, along with another well-known identity:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1),}</annotation>

</semantics>

</math></span></span>

produces the following derivation:<sup id="cite_ref-FOOTNOTEWheatstone1854_5-0" class="reference"><a href="#cite_note-FOOTNOTEWheatstone1854-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mrow>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mrow>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mrow>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</munder>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munder>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=1}^{n}k^{3}&=1+8+27+64+\cdots +n^{3}\\&=\underbrace {1} _{1^{3}}+\underbrace {3+5} _{2^{3}}+\underbrace {7+9+11} _{3^{3}}+\underbrace {13+15+17+19} _{4^{3}}+\cdots +\underbrace {\left(n^{2}-n+1\right)+\cdots +\left(n^{2}+n-1\right)} _{n^{3}}\\&=\underbrace {\underbrace {\underbrace {\underbrace {1} _{1^{2}}+3} _{2^{2}}+5} _{3^{2}}+\cdots +\left(n^{2}+n-1\right)} _{\left({\frac {n^{2}+n}{2}}\right)^{2}}\\&=(1+2+\cdots +n)^{2}\\&=\left(\sum _{k=1}^{n}k\right)^{2}.\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p><a href="#CITEREFRow1893">Row (1893)</a> obtains another proof by summing the numbers in a square <a href="Multiplication_table" title="Multiplication table">multiplication table</a> in two different ways. The sum of the <span class="texhtml mvar" style="font-style:italic;">i</span>th row is <span class="texhtml mvar" style="font-style:italic;">i</span> times a triangular number, from which it follows that the sum of all the rows is the square of a triangular number. Alternatively, one can decompose the table into a sequence of nested <a href="Gnomon_(figure)" title="Gnomon (figure)">gnomons</a>, each consisting of the products in which the larger of the two terms is some fixed value. The sum within each gnomon is a cube, so the sum of the whole table is a sum of cubes.<sup id="cite_ref-FOOTNOTERow1893_6-0" class="reference"><a href="#cite_note-FOOTNOTERow1893-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<p>In the more recent mathematical literature, <a href="#CITEREFEdmonds1957">Edmonds (1957)</a> provides a proof using <a href="Summation_by_parts" title="Summation by parts">summation by parts</a>.<sup id="cite_ref-FOOTNOTEEdmonds1957_7-0" class="reference"><a href="#cite_note-FOOTNOTEEdmonds1957-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFStein1971">Stein (1971)</a> uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Stein observes that it may also be proved easily (but uninformatively) by induction, and states that <a href="#CITEREFToeplitz1963">Toeplitz (1963)</a> provides "an interesting old Arabic proof".<sup id="cite_ref-FOOTNOTEStein1971_4-1" class="reference"><a href="#cite_note-FOOTNOTEStein1971-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFKanim2004">Kanim (2004)</a> provides a purely visual proof,<sup id="cite_ref-FOOTNOTEKanim2004_9-0" class="reference"><a href="#cite_note-FOOTNOTEKanim2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFBenjaminOrrison2002">Benjamin & Orrison (2002)</a> provide two additional proofs,<sup id="cite_ref-FOOTNOTEBenjaminOrrison2002_10-0" class="reference"><a href="#cite_note-FOOTNOTEBenjaminOrrison2002-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and <a href="#CITEREFNelsen1993">Nelsen (1993)</a> gives seven geometric proofs.<sup id="cite_ref-FOOTNOTENelsen1993_11-0" class="reference"><a href="#cite_note-FOOTNOTENelsen1993-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>A similar result to Nicomachus's theorem holds for all <a href="Faulhaber's_formula" title="Faulhaber's formula">power sums</a>, namely that odd power sums (sums of odd powers) are a polynomial in triangular numbers.

These are called <a href="Faulhaber's_formula#Faulhaber_polynomials" title="Faulhaber's formula">Faulhaber polynomials</a>, of which the sum of cubes is the simplest and most elegant example.

However, in no other case is one power sum a square of another.<sup id="cite_ref-FOOTNOTEEdmonds1957_7-1" class="reference"><a href="#cite_note-FOOTNOTEEdmonds1957-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p><a href="#CITEREFStroeker1995">Stroeker (1995)</a> studies more general conditions under which the sum of a consecutive sequence of cubes forms a square.<sup id="cite_ref-FOOTNOTEStroeker1995_1-1" class="reference"><a href="#cite_note-FOOTNOTEStroeker1995-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFGarrettHummel2004">Garrett & Hummel (2004)</a> and <a href="#CITEREFWarnaar2004">Warnaar (2004)</a> study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

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

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-FOOTNOTEStroeker1995-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEStroeker1995_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEStroeker1995_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFStroeker1995">Stroeker (1995)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEPengelley2002-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPengelley2002_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPengelley2002">Pengelley (2002)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEBressoud2004-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBressoud2004_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBressoud2004">Bressoud (2004)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEStein1971-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEStein1971_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEStein1971_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFStein1971">Stein (1971)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEWheatstone1854-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWheatstone1854_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWheatstone1854">Wheatstone (1854)</a>.</span>

</li>

</li>

<li id="cite_note-FOOTNOTEEdmonds1957-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEEdmonds1957_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEEdmonds1957_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEdmonds1957">Edmonds (1957)</a>.</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFStein1971">Stein (1971)</a>; see also <a href="#CITEREFBenjaminQuinnWurtz2006">Benjamin, Quinn & Wurtz 2006</a></span>

</li>

<li id="cite_note-FOOTNOTEKanim2004-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKanim2004_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKanim2004">Kanim (2004)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEBenjaminOrrison2002-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBenjaminOrrison2002_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBenjaminOrrison2002">Benjamin & Orrison (2002)</a>.</span>

</li>

<li id="cite_note-FOOTNOTENelsen1993-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENelsen1993_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNelsen1993">Nelsen (1993)</a>.</span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFGarrettHummel2004">Garrett & Hummel (2004)</a>; <a href="#CITEREFWarnaar2004">Warnaar (2004)</a></span>

</li>

</ol></div>

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

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</style><cite id="CITEREFBenjaminOrrison2002" class="citation cs2"><a href="Arthur_T._Benjamin" title="Arthur T. Benjamin">Benjamin, Arthur T.</a>; Orrison, M. E. (2002), <a rel="nofollow" class="external text" href="http://www.math.hmc.edu/~orrison/research/papers/two_quick.pdf">"Two quick combinatorial proofs of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum k^{3}={n+1 \choose 2}^{2}}">

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</math></span><img src="./5bbffa1094d4827a5e51f6b06d79e83be429eddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.476ex; height:3.843ex;" alt="{\displaystyle \textstyle \sum k^{3}={n+1 \choose 2}^{2}}" loading="lazy"></span>"</a> <span class="cs1-format">(PDF)</span>, <i><a href="College_Mathematics_Journal" class="mw-redirect" title="College Mathematics Journal">College Mathematics Journal</a></i>, <b>33</b> (5): <span class="nowrap">406–</span>408, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1559017">10.2307/1559017</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/1559017">1559017</a></cite>.</li>

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<li><cite id="CITEREFGarrettHummel2004" class="citation cs2">Garrett, Kristina C.; Hummel, Kristen (2004), <a rel="nofollow" class="external text" href="http://www.combinatorics.org/Volume_11/Abstracts/v11i1r9.html">"A combinatorial proof of the sum of <span class="texhtml mvar" style="font-style:italic;">q</span>-cubes"</a>, <i><a href="Electronic_Journal_of_Combinatorics" title="Electronic Journal of Combinatorics">Electronic Journal of Combinatorics</a></i>, <b>11</b> (1), Research Paper 9, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.37236%2F1762">10.37236/1762</a></span>, <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=2034423">2034423</a></cite>.</li>

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<li><cite id="CITEREFKanim2004" class="citation cs2">Kanim, Katherine (2004), "Proofs without words: The sum of cubes—An extension of Archimedes' sum of squares", <i><a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>77</b> (4): <span class="nowrap">298–</span>299, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3219288">10.2307/3219288</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/3219288">3219288</a></cite>.</li>

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<li><cite id="CITEREFRow1893" class="citation cs2">Row, T. Sundara (1893), <a href="Geometric_Exercises_in_Paper_Folding" title="Geometric Exercises in Paper Folding"><i>Geometric Exercises in Paper Folding</i></a>, Madras: Addison, <a rel="nofollow" class="external text" href="https://archive.org/details/geometricexerci00raogoog/page/n61/mode/2up">pp. 47–48</a></cite>.</li>

<li><cite id="CITEREFStein1971" class="citation cs2">Stein, Robert G. (1971), "A combinatorial proof that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum k^{3}=(\sum k)^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum k^{3}=(\sum k)^{2}}</annotation>

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</math></span><img src="./dbd77136a26453f22b14dd73bdc068fe8adc81cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.121ex; height:3.009ex;" alt="{\displaystyle \textstyle \sum k^{3}=(\sum k)^{2}}" loading="lazy"></span>", <i><a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>44</b> (3): <span class="nowrap">161–</span>162, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2688231">10.2307/2688231</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/2688231">2688231</a></cite>.</li>

<li><cite id="CITEREFStroeker1995" class="citation cs2">Stroeker, R. J. (1995), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=CM_1995__97_1-2_295_0">"On the sum of consecutive cubes being a perfect square"</a>, <i><a href="Compositio_Mathematica" title="Compositio Mathematica">Compositio Mathematica</a></i>, <b>97</b> (<span class="nowrap">1–</span>2): <span class="nowrap">295–</span>307, <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=1355130">1355130</a></cite>.</li>

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<li><cite id="CITEREFWheatstone1854" class="citation cs2"><a href="Charles_Wheatstone" title="Charles Wheatstone">Wheatstone, C.</a> (1854), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1432033">"On the formation of powers from arithmetical progressions"</a>, <i><a href="Proceedings_of_the_Royal_Society_of_London" class="mw-redirect" title="Proceedings of the Royal Society of London">Proceedings of the Royal Society of London</a></i>, <b>7</b>: <span class="nowrap">145–</span>151, <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/1854RSPS....7..145W">1854RSPS....7..145W</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspl.1854.0036">10.1098/rspl.1854.0036</a></span></cite>.</li></ul>

</div>

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

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Nicomachus's_theorem"><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/NicomachussTheorem.html">"Nicomachus's theorem"</a>, <i><a href="MathWorld" title="MathWorld">MathWorld</a></i></cite></span></li>

<li><a rel="nofollow" class="external text" href="http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32">A visual proof of Nicomachus's theorem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191019215037/http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32">Archived</a> 2019-10-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>

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

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

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

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

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

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

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

<li><a href="Star_number" title="Star number">Star numbers</a></li></ul>

</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 numbers</a></li>

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

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

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

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

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

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

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

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

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

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

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

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral numbers</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="Cube_(algebra)" title="Cube (algebra)">Cube numbers</a></li>

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

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

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

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula numbers</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="Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></li>

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

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

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher <a href="Dimension" title="Dimension">dimensional</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 id="non-centered12" scope="row" class="navbox-group" style="width:1%">non-centered</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="Fifth_power_(algebra)" title="Fifth power (algebra)">5-hypercube numbers</a></li>

<li><a href="Sixth_power" title="Sixth power">6-hypercube numbers</a></li>

<li><a href="Seventh_power" title="Seventh power">7-hypercube numbers</a></li>

<li><a href="Eighth_power" title="Eighth power">8-hypercube numbers</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers743" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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 <i>a</i> × 2<sup><i>b</i></sup> ± 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="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li>

<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="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> <br>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><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-06-22" href="https://en.wikipedia.org/wiki/?title=Squared_triangular_number&oldid=1296888607">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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