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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Almost integer</span></span>
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<p>In <a href="Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>, an <b>almost integer</b> (or <b>near-integer</b>) is any number that is not an <a href="Integer" title="Integer">integer</a> but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Almost_integers_relating_to_the_golden_ratio_and_Fibonacci_numbers">Almost integers relating to the golden ratio and Fibonacci numbers</h2></div>
<p>Some examples of almost integers are high powers of the <a href="Golden_ratio" title="Golden ratio">golden ratio</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 \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>1.618</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}</annotation>
</semantics>
</math></span><img src="./c15a855b1a1b214e2791a266af9396e70c163fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.816ex; height:5.843ex;" alt="{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}" loading="lazy"></span>, for example:
</p>
<dl><dd><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 {\begin{aligned}\phi ^{17}&={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\\[6pt]\phi ^{18}&=2889+1292{\sqrt {5}}\approx 5777.999827\\[6pt]\phi ^{19}&={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>17</mn>
</mrow>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>3571</mn>
<mo>+</mo>
<mn>1597</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>3571.00028</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>18</mn>
</mrow>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>2889</mn>
<mo>+</mo>
<mn>1292</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>5777.999827</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>19</mn>
</mrow>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>9349</mn>
<mo>+</mo>
<mn>4181</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>9349.000107</mn>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\phi ^{17}&={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\\[6pt]\phi ^{18}&=2889+1292{\sqrt {5}}\approx 5777.999827\\[6pt]\phi ^{19}&={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./517ac892a0426835ff7e4378faf421aeef1d9fdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:38.556ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}\phi ^{17}&={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\\[6pt]\phi ^{18}&=2889+1292{\sqrt {5}}\approx 5777.999827\\[6pt]\phi ^{19}&={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>The fact that these powers approach integers is non-coincidental, because the golden ratio is a <a href="Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a>.
</p><p>The ratios of <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci</a> or <a href="Lucas_number" title="Lucas number">Lucas</a> numbers can also make almost integers, for instance:
</p>
<ul><li><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 {\frac {\operatorname {Fib} (360)}{\operatorname {Fib} (216)}}\approx 1242282009792667284144565908481.999999999999999999999999999999195}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>Fib</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>360</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>Fib</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>216</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>1242282009792667284144565908481.999999999999999999999999999999195</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Fib} (360)}{\operatorname {Fib} (216)}}\approx 1242282009792667284144565908481.999999999999999999999999999999195}</annotation>
</semantics>
</math></span><img src="./d5b4d836d3f3b136db2fcff075a8133962340b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:87.732ex; height:6.509ex;" alt="{\displaystyle {\frac {\operatorname {Fib} (360)}{\operatorname {Fib} (216)}}\approx 1242282009792667284144565908481.999999999999999999999999999999195}" loading="lazy"></span></li>
<li><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 {\frac {\operatorname {Lucas} (361)}{\operatorname {Lucas} (216)}}\approx 2010054515457065378082322433761.000000000000000000000000000000497}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>361</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>216</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>2010054515457065378082322433761.000000000000000000000000000000497</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Lucas} (361)}{\operatorname {Lucas} (216)}}\approx 2010054515457065378082322433761.000000000000000000000000000000497}</annotation>
</semantics>
</math></span><img src="./818979a02611e0f0139089b60153927b10f09cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:90.132ex; height:6.509ex;" alt="{\displaystyle {\frac {\operatorname {Lucas} (361)}{\operatorname {Lucas} (216)}}\approx 2010054515457065378082322433761.000000000000000000000000000000497}" loading="lazy"></span></li></ul>
<p>The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:
</p>
<ul><li><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 a(n)={\frac {\operatorname {Fib} (45\times 2^{n})}{\operatorname {Fib} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>Fib</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>45</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>Fib</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>27</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>18</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a(n)={\frac {\operatorname {Fib} (45\times 2^{n})}{\operatorname {Fib} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n})}</annotation>
</semantics>
</math></span><img src="./5c575ead6bc0ade13c4d88483823e6d64c757e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.492ex; height:6.509ex;" alt="{\displaystyle a(n)={\frac {\operatorname {Fib} (45\times 2^{n})}{\operatorname {Fib} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n})}" loading="lazy"></span></li>
<li><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 a(n)={\frac {\operatorname {Lucas} (45\times 2^{n}+1)}{\operatorname {Lucas} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n}+1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>45</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>27</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mi>Lucas</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>18</mn>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a(n)={\frac {\operatorname {Lucas} (45\times 2^{n}+1)}{\operatorname {Lucas} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n}+1)}</annotation>
</semantics>
</math></span><img src="./b8c8bee3b6b591a0005162ebef1cfc2bde7505b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.896ex; height:6.509ex;" alt="{\displaystyle a(n)={\frac {\operatorname {Lucas} (45\times 2^{n}+1)}{\operatorname {Lucas} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n}+1)}" loading="lazy"></span></li></ul>
<p>As <i>n</i> increases, the number of consecutive nines or zeros beginning at the tenths place of <i>a</i>(<i>n</i>) approaches infinity.
</p>
<div class="mw-heading mw-heading2"><h2 id="Almost_integers_relating_to_e_and_π">Almost integers relating to <i>e</i> and <span class="texhtml mvar" style="font-style:italic;">π</span></h2></div>
<p>Other occurrences of non-coincidental near-integers involve the three largest <a href="Heegner_number" title="Heegner number">Heegner numbers</a>:
</p>
<ul><li><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 e^{\pi {\sqrt {43}}}\approx 884736743.999777466}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>43</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mn>884736743.999777466</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {43}}}\approx 884736743.999777466}</annotation>
</semantics>
</math></span><img src="./91292816b8cd5afaf8e5a25f867434aa498540d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.94ex; height:3.009ex;" alt="{\displaystyle e^{\pi {\sqrt {43}}}\approx 884736743.999777466}" loading="lazy"></span></li>
<li><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 e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>67</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mn>147197952743.999998662454</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454}</annotation>
</semantics>
</math></span><img src="./182b3d67b5dab16e1bb82e9f10f7940dc9fbbbf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:36.915ex; height:3.009ex;" alt="{\displaystyle e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454}" loading="lazy"></span></li>
<li><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 e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>163</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mn>262537412640768743.99999999999925007</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007}</annotation>
</semantics>
</math></span><img src="./691a48bf9a680ceef7192be46c21c0f88e7de455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:50.524ex; height:3.009ex;" alt="{\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007}" loading="lazy"></span></li></ul>
<p>where the non-coincidence can be better appreciated when expressed in the common simple form:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><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 e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-(2.225\ldots )\times 10^{-4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>43</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mn>12</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mn>9</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>744</mn>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mn>2.225</mn>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-(2.225\ldots )\times 10^{-4}}</annotation>
</semantics>
</math></span><img src="./062411788a9a1f0f149447a3763105085a4a63f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.714ex; height:3.509ex;" alt="{\displaystyle e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-(2.225\ldots )\times 10^{-4}}" loading="lazy"></span></dd>
<dd><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 e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-(1.337\ldots )\times 10^{-6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>67</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mn>12</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mn>21</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>744</mn>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mn>1.337</mn>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>6</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-(1.337\ldots )\times 10^{-6}}</annotation>
</semantics>
</math></span><img src="./118000e7383acbd81b6270d344b658e1731c9991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.876ex; height:3.509ex;" alt="{\displaystyle e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-(1.337\ldots )\times 10^{-6}}" loading="lazy"></span></dd>
<dd><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 e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-(7.499\ldots )\times 10^{-13}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>163</mn>
</msqrt>
</mrow>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mn>12</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mn>231</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>744</mn>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mn>7.499</mn>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>13</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-(7.499\ldots )\times 10^{-13}}</annotation>
</semantics>
</math></span><img src="./6a591a15877d8684037cfdc6089c357f792d6d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.683ex; height:3.509ex;" alt="{\displaystyle e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-(7.499\ldots )\times 10^{-13}}" loading="lazy"></span></dd></dl>
<p>where
</p>
<dl><dd><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 21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>21</mn>
<mo>=</mo>
<mn>3</mn>
<mo>×<!-- × --></mo>
<mn>7</mn>
<mo>,</mo>
<mspace width="1em"></mspace>
<mn>231</mn>
<mo>=</mo>
<mn>3</mn>
<mo>×<!-- × --></mo>
<mn>7</mn>
<mo>×<!-- × --></mo>
<mn>11</mn>
<mo>,</mo>
<mspace width="1em"></mspace>
<mn>744</mn>
<mo>=</mo>
<mn>24</mn>
<mo>×<!-- × --></mo>
<mn>31</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31}</annotation>
</semantics>
</math></span><img src="./5d5fbb0d3bbb429d67b84ff7331f225bf21e7f49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.294ex; height:2.509ex;" alt="{\displaystyle 21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31}" loading="lazy"></span></dd></dl>
<p>and the reason for the squares is due to certain <a href="Eisenstein_series" title="Eisenstein series">Eisenstein series</a>. The constant <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 e^{\pi {\sqrt {163}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>163</mn>
</msqrt>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi {\sqrt {163}}}}</annotation>
</semantics>
</math></span><img src="./74fd61cac30067a5b97427918bbc23b1f310115f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.092ex; height:3.009ex;" alt="{\displaystyle e^{\pi {\sqrt {163}}}}" loading="lazy"></span>
is sometimes referred to as <a href="Ramanujan's_constant" class="mw-redirect" title="Ramanujan's constant">Ramanujan's constant</a>.
</p><p>Almost integers that involve the mathematical constants <a href="Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> and <a href="E_(mathematical_constant)" title="E (mathematical constant)">e</a> have often puzzled mathematicians. An example 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 e^{\pi }-\pi =19.999099979189\ldots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<mo>=</mo>
<mn>19.999099979189</mn>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }-\pi =19.999099979189\ldots }</annotation>
</semantics>
</math></span><img src="./eeb1c4eb1b2eca4ba09ee9c53bd7efa4f39a5dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.56ex; height:2.509ex;" alt="{\displaystyle e^{\pi }-\pi =19.999099979189\ldots }" loading="lazy"></span>
The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to <a href="Jacobi_theta_functions" class="mw-redirect" title="Jacobi theta functions">Jacobi theta functions</a> as follows:
<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 \sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{-\pi k^{2}}=1.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<mn>8</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<mo>=</mo>
<mn>1.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{-\pi k^{2}}=1.}</annotation>
</semantics>
</math></span></span>
The first term dominates since the sum of the terms for <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 k\geq 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation>
</semantics>
</math></span><img src="./c797a67c0a51167d373c013a9a020f4568a11754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}" loading="lazy"></span> total <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 \sim 0.0003436.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∼<!-- ∼ --></mo>
<mn>0.0003436.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sim 0.0003436.}</annotation>
</semantics>
</math></span><img src="./75dfe1a0b4a7708e21f7fce15c475fb72906196f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.047ex; height:2.176ex;" alt="{\displaystyle \sim 0.0003436.}" loading="lazy"></span> The sum can therefore be truncated 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 \left(8\pi -2\right)e^{-\pi }\approx 1,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>(</mo>
<mrow>
<mn>8</mn>
<mi>π<!-- π --></mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mn>1</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left(8\pi -2\right)e^{-\pi }\approx 1,}</annotation>
</semantics>
</math></span><img src="./a7c49899f486d86d0bf2e7bef4dcce188ba313e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.138ex; height:3.009ex;" alt="{\displaystyle \left(8\pi -2\right)e^{-\pi }\approx 1,}" loading="lazy"></span>
where solving for <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 e^{\pi }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }}</annotation>
</semantics>
</math></span><img src="./eefb10ad1f3612be16a802dd913a9edfb5b9d823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.258ex; height:2.343ex;" alt="{\displaystyle e^{\pi }}" loading="lazy"></span> gives <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 e^{\pi }\approx 8\pi -2.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mn>8</mn>
<mi>π<!-- π --></mi>
<mo>−<!-- − --></mo>
<mn>2.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }\approx 8\pi -2.}</annotation>
</semantics>
</math></span><img src="./e6df25f2a2466ece1d53813563d59d74f2f9122e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.5ex; height:2.509ex;" alt="{\displaystyle e^{\pi }\approx 8\pi -2.}" loading="lazy"></span>
Rewriting the approximation for <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 e^{\pi }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }}</annotation>
</semantics>
</math></span><img src="./eefb10ad1f3612be16a802dd913a9edfb5b9d823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.258ex; height:2.343ex;" alt="{\displaystyle e^{\pi }}" loading="lazy"></span> and using the approximation for <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 7\pi \approx 22}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>7</mn>
<mi>π<!-- π --></mi>
<mo>≈<!-- ≈ --></mo>
<mn>22</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 7\pi \approx 22}</annotation>
</semantics>
</math></span><img src="./ded50d84a3e78827236b6a06df6f5f94822b2c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.918ex; height:2.176ex;" alt="{\displaystyle 7\pi \approx 22}" loading="lazy"></span> gives
<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 e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>≈<!-- ≈ --></mo>
<mi>π<!-- π --></mi>
<mo>+</mo>
<mn>7</mn>
<mi>π<!-- π --></mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>≈<!-- ≈ --></mo>
<mi>π<!-- π --></mi>
<mo>+</mo>
<mn>22</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>=</mo>
<mi>π<!-- π --></mi>
<mo>+</mo>
<mn>20.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.}</annotation>
</semantics>
</math></span></span>
Thus, rearranging terms gives <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 e^{\pi }-\pi \approx 20.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>π<!-- π --></mi>
<mo>≈<!-- ≈ --></mo>
<mn>20.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{\pi }-\pi \approx 20.}</annotation>
</semantics>
</math></span><img src="./9bff911f13632f62576fd1a8d3ead269208f160a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.5ex; height:2.509ex;" alt="{\displaystyle e^{\pi }-\pi \approx 20.}" loading="lazy"></span> Ironically, the crude approximation for <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 7\pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>7</mn>
<mi>π<!-- π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 7\pi }</annotation>
</semantics>
</math></span><img src="./e208cc9deccf34aac0a1c5ba93798dace5a9a11a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 7\pi }" loading="lazy"></span> yields an additional order of magnitude of precision.
<sup id="cite_ref-MathWorld_1-1" class="reference"><a href="#cite_note-MathWorld-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Another example involving these constants 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 e+\pi +e\pi +e^{\pi }+\pi ^{e}=59.9994590558\ldots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>e</mi>
<mo>+</mo>
<mi>π<!-- π --></mi>
<mo>+</mo>
<mi>e</mi>
<mi>π<!-- π --></mi>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>π<!-- π --></mi>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
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<mo>=</mo>
<mn>59.9994590558</mn>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e+\pi +e\pi +e^{\pi }+\pi ^{e}=59.9994590558\ldots }</annotation>
</semantics>
</math></span><img src="./4d9b5526aa55db8c7c78161ba5cc091c4adda0ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:41.588ex; height:2.509ex;" alt="{\displaystyle e+\pi +e\pi +e^{\pi }+\pi ^{e}=59.9994590558\ldots }" loading="lazy"></span>
</p><p><br>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Schizophrenic_number" title="Schizophrenic number">Schizophrenic number</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-MathWorld-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-MathWorld_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MathWorld_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="Eric_Weisstein" class="mw-redirect" title="Eric Weisstein">Eric Weisstein</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/AlmostInteger.html">"Almost Integer"</a> at <a href="MathWorld" title="MathWorld">MathWorld</a></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#">"More on e^(pi*SQRT(163))"</a>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf">J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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