imported>Boonerquad: Clarified confusing language.

2025-03-10T21:36:15Z

Clarified confusing language.

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{{short description|Any number that is not an integer but is very close to one}}
[[File:Almost integer in triangle.svg|thumb|250px|[[Ed Pegg Jr.]] noted that the length ''d'' equals <math>\frac{1}{2}\sqrt{\frac{1}{30}(61421-23\sqrt{5831385})} </math>, which is very close to 7 (approximately 7.0000000857)<ref name="MathWorld"/>]]

In [[recreational mathematics]], an '''almost integer''' (or '''near-integer''') is any number that is not an [[integer]] but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected.

== Almost integers relating to the golden ratio and Fibonacci numbers==

Some examples of almost integers are high powers of the [[golden ratio]] <math>\phi=\frac{1+\sqrt5}{2}\approx 1.618</math>, for example:

: <math>
\begin{align}
\phi^{17} & =\frac{3571+1597\sqrt5}{2}\approx 3571.00028 \\[6pt]
\phi^{18} & =2889+1292\sqrt5 \approx 5777.999827 \\[6pt]
\phi^{19} & =\frac{9349+4181\sqrt5}{2}\approx 9349.000107
\end{align}
</math>

The fact that these powers approach integers is non-coincidental, because the golden ratio is a [[Pisot–Vijayaraghavan number]].

The ratios of [[Fibonacci_number|Fibonacci]] or [[Lucas_number|Lucas]] numbers can also make almost integers, for instance:

* <math> \frac{\operatorname{Fib}(360)}{\operatorname{Fib}(216)} \approx 1242282009792667284144565908481.999999999999999999999999999999195 </math>
* <math> \frac{\operatorname{Lucas}(361)}{\operatorname{Lucas}(216)} \approx 2010054515457065378082322433761.000000000000000000000000000000497
</math>

The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:

* <math>a(n) = \frac{\operatorname{Fib}(45\times2^n)}{\operatorname{Fib}(27\times2^n)} \approx \operatorname{Lucas}(18\times2^n)</math>
* <math>a(n) = \frac{\operatorname{Lucas}(45\times2^n+1)}{\operatorname{Lucas}(27\times2^n)} \approx \operatorname{Lucas}(18\times2^n+1)</math>

As ''n'' increases, the number of consecutive nines or zeros beginning at the tenths place of ''a''(''n'') approaches infinity.

== Almost integers relating to ''e'' and {{pi}}==

Other occurrences of non-coincidental near-integers involve the three largest [[Heegner number]]s:
* <math>e^{\pi\sqrt{43}}\approx 884736743.999777466</math>
* <math>e^{\pi\sqrt{67}}\approx 147197952743.999998662454</math>
* <math>e^{\pi\sqrt{163}}\approx 262537412640768743.99999999999925007</math>
where the non-coincidence can be better appreciated when expressed in the common simple form:<ref>{{Cite web|url=http://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#|title=More on e^(pi*SQRT(163))}}</ref>
:<math>e^{\pi\sqrt{43}}=12^3(9^2-1)^3+744-(2.225\ldots)\times 10^{-4}</math>
:<math>e^{\pi\sqrt{67}}=12^3(21^2-1)^3+744-(1.337\ldots)\times 10^{-6}</math>
:<math>e^{\pi\sqrt{163}}=12^3(231^2-1)^3+744-(7.499\ldots)\times 10^{-13}</math>
where
:<math>21=3\times7, \quad 231=3\times7\times11, \quad 744=24\times 31</math>
and the reason for the squares is due to certain [[Eisenstein series]]. The constant <math>e^{\pi\sqrt{163}}</math>
is sometimes referred to as [[Ramanujan's constant]].

Almost integers that involve the mathematical constants [[pi|{{pi}}]] and [[E (mathematical constant)|e]] have often puzzled mathematicians. An example is: <math>e^\pi-\pi=19.999099979189\ldots</math>
The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to [[Jacobi theta functions]] as follows:
<math display=block>\sum_{k=1}^{\infty}\left( 8\pi k^2 -2 \right) e^{-\pi k^2} = 1.</math>
The first term dominates since the sum of the terms for <math>k\geq 2</math> total <math>\sim 0.0003436.</math> The sum can therefore be truncated to
<math>\left( 8\pi -2\right) e^{-\pi}\approx 1,</math>
where solving for <math>e^{\pi}</math> gives <math>e^{\pi} \approx 8\pi -2.</math>
Rewriting the approximation for <math>e^{\pi}</math> and using the approximation for <math>7\pi \approx 22</math> gives
<math display=block> e^{\pi} \approx \pi + 7\pi - 2 \approx \pi + 22-2 = \pi+20.</math>
Thus, rearranging terms gives <math>e^{\pi} - \pi \approx 20.</math> Ironically, the crude approximation for <math>7\pi</math> yields an additional order of magnitude of precision.
<ref name="MathWorld">[[Eric Weisstein]], [http://mathworld.wolfram.com/AlmostInteger.html "Almost Integer"] at [[MathWorld]]</ref>

Another example involving these constants is: <math>e+\pi+e\pi+e^\pi+\pi^e=59.9994590558\ldots</math>

==See also==
*[[Schizophrenic number]]

== References ==
{{Reflist}}

== External links ==
* [http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought]

[[Category:Integers]]
[[Category:Recreational mathematics]]

Almost integer - Revision history

imported>Boonerquad: Clarified confusing language.