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Almost integer
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<h2 id="almost-integers-relating-to-the-golden-ratio-and-fibonacci-numbers">Almost integers relating to the golden ratio and Fibonacci numbers</h2> <p>Some examples of almost integers are high powers of the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> ϕ = 1 + 5 2 ≈ 1.618 {\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618} , for example: </p> ϕ 17 = 3571 + 1597 5 2 ≈ 3571.00028 ϕ 18 = 2889 + 1292 5 ≈ 5777.999827 ϕ 19 = 9349 + 4181 5 2 ≈ 9349.000107 {\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}}} <p>The fact that these powers approach integers is non-coincidental, because the golden ratio is a <a href="/facts/Pisot%E2%80%93Vijayaraghavan_number/bkCgwe0Q">Pisot–Vijayaraghavan number</a>. </p><p>The ratios of <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci</a> or <a href="/facts/Lucas_number/tVbP4yQr">Lucas</a> numbers can also make almost integers, for instance: </p> <ul><li> Fib ( 360 ) Fib ( 216 ) ≈ 1242282009792667284144565908481.999999999999999999999999999999195 {\displaystyle {\frac {\operatorname {Fib} (360)}{\operatorname {Fib} (216)}}\approx 1242282009792667284144565908481.999999999999999999999999999999195} </li> <li> Lucas ( 361 ) Lucas ( 216 ) ≈ 2010054515457065378082322433761.000000000000000000000000000000497 {\displaystyle {\frac {\operatorname {Lucas} (361)}{\operatorname {Lucas} (216)}}\approx 2010054515457065378082322433761.000000000000000000000000000000497} </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> a ( n ) = Fib ( 45 × 2 n ) Fib ( 27 × 2 n ) ≈ Lucas ( 18 × 2 n ) {\displaystyle a(n)={\frac {\operatorname {Fib} (45\times 2^{n})}{\operatorname {Fib} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n})} </li> <li> a ( n ) = Lucas ( 45 × 2 n + 1 ) Lucas ( 27 × 2 n ) ≈ Lucas ( 18 × 2 n + 1 ) {\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)} </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> <h2 id="almost-integers-relating-to-eand-">Almost integers relating to <i>e</i> and π</h2> <p>Other occurrences of non-coincidental near-integers involve the three largest <a href="/facts/Heegner_number/ExPvl9UM">Heegner numbers</a>: </p> <ul><li> e π 43 ≈ 884736743.999777466 {\displaystyle e^{\pi {\sqrt {43}}}\approx 884736743.999777466} </li> <li> e π 67 ≈ 147197952743.999998662454 {\displaystyle e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454} </li> <li> e π 163 ≈ 262537412640768743.99999999999925007 {\displaystyle e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007} </li></ul> <p>where the non-coincidence can be better appreciated when expressed in the common simple form:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> e π 43 = 12 3 ( 9 2 − 1 ) 3 + 744 − ( 2.225 … ) × 10 − 4 {\displaystyle e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-(2.225\ldots )\times 10^{-4}} e π 67 = 12 3 ( 21 2 − 1 ) 3 + 744 − ( 1.337 … ) × 10 − 6 {\displaystyle e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-(1.337\ldots )\times 10^{-6}} e π 163 = 12 3 ( 231 2 − 1 ) 3 + 744 − ( 7.499 … ) × 10 − 13 {\displaystyle e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-(7.499\ldots )\times 10^{-13}} <p>where </p> 21 = 3 × 7 , 231 = 3 × 7 × 11 , 744 = 24 × 31 {\displaystyle 21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31} <p>and the reason for the squares is due to certain <a href="/facts/Eisenstein_series/23vgJLFl">Eisenstein series</a>. The constant e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} is sometimes referred to as <a href="/facts/Ramanujan%27s_constant/ExPvl9UM">Ramanujan's constant</a>. </p><p>Almost integers that involve the mathematical constants <a href="/facts/Pi/vC0UBj1q">π</a> and <a href="/facts/E_(mathematical_constant)/coEtSBk8">e</a> have often puzzled mathematicians. An example is: e π − π = 19.999099979189 … {\displaystyle e^{\pi }-\pi =19.999099979189\ldots } 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="/facts/Jacobi_theta_functions/g09xvoQQ">Jacobi theta functions</a> as follows: ∑ k = 1 ∞ ( 8 π k 2 − 2 ) e − π k 2 = 1. {\displaystyle \sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{-\pi k^{2}}=1.} The first term dominates since the sum of the terms for k ≥ 2 {\displaystyle k\geq 2} total ∼ 0.0003436. {\displaystyle \sim 0.0003436.} The sum can therefore be truncated to ( 8 π − 2 ) e − π ≈ 1 , {\displaystyle \left(8\pi -2\right)e^{-\pi }\approx 1,} where solving for e π {\displaystyle e^{\pi }} gives e π ≈ 8 π − 2. {\displaystyle e^{\pi }\approx 8\pi -2.} Rewriting the approximation for e π {\displaystyle e^{\pi }} and using the approximation for 7 π ≈ 22 {\displaystyle 7\pi \approx 22} gives e π ≈ π + 7 π − 2 ≈ π + 22 − 2 = π + 20. {\displaystyle e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.} Thus, rearranging terms gives e π − π ≈ 20. {\displaystyle e^{\pi }-\pi \approx 20.} Ironically, the crude approximation for 7 π {\displaystyle 7\pi } yields an additional order of magnitude of precision. <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>Another example involving these constants is: e + π + e π + e π + π e = 59.9994590558 … {\displaystyle e+\pi +e\pi +e^{\pi }+\pi ^{e}=59.9994590558\ldots } </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Schizophrenic_number/KLcGGhdX">Schizophrenic number</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a 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> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"More on e^(pi*SQRT(163))". <a href="https://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en" target="_blank">https://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Eric Weisstein, "Almost Integer" at MathWorld <a href="/wiki/Eric_Weisstein" target="_blank">/wiki/Eric_Weisstein</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>