# Almost integer

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 $\phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618$, for example:
    ${\begin{aligned}\phi ^{17}&={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\\[6pt]\phi ^{18}&=2889+1292{\sqrt {5}}\approx 5777.999827\\[6pt]\phi ^{19}&={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\end{aligned}}$
The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number.
The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance:
* ${\frac {\operatorname {Fib} (360)}{\operatorname {Fib} (216)}}\approx 1242282009792667284144565908481.999999999999999999999999999999195$
* ${\frac {\operatorname {Lucas} (361)}{\operatorname {Lucas} (216)}}\approx 2010054515457065378082322433761.000000000000000000000000000000497$
The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:
* $a(n)={\frac {\operatorname {Fib} (45\times 2^{n})}{\operatorname {Fib} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n})$
* $a(n)={\frac {\operatorname {Lucas} (45\times 2^{n}+1)}{\operatorname {Lucas} (27\times 2^{n})}}\approx \operatorname {Lucas} (18\times 2^{n}+1)$
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 π

Other occurrences of non-coincidental near-integers involve the three largest Heegner numbers:
* $e^{\pi {\sqrt {43}}}\approx 884736743.999777466$
* $e^{\pi {\sqrt {67}}}\approx 147197952743.999998662454$
* $e^{\pi {\sqrt {163}}}\approx 262537412640768743.99999999999925007$
where the non-coincidence can be better appreciated when expressed in the common simple form:
    $e^{\pi {\sqrt {43}}}=12^{3}(9^{2}-1)^{3}+744-(2.225\ldots )\times 10^{-4}$
    $e^{\pi {\sqrt {67}}}=12^{3}(21^{2}-1)^{3}+744-(1.337\ldots )\times 10^{-6}$
    $e^{\pi {\sqrt {163}}}=12^{3}(231^{2}-1)^{3}+744-(7.499\ldots )\times 10^{-13}$
where
    $21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31$
and the reason for the squares is due to certain Eisenstein series. The constant $e^{\pi {\sqrt {163}}}$ is sometimes referred to as Ramanujan's constant.
Almost integers that involve the mathematical constants π and e have often puzzled mathematicians. An example is: $e^{\pi }-\pi =19.999099979189\ldots$ The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: $\sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{-\pi k^{2}}=1.$ The first term dominates since the sum of the terms for $k\geq 2$ total $\sim 0.0003436.$ The sum can therefore be truncated to $\left(8\pi -2\right)e^{-\pi }\approx 1,$ where solving for $e^{\pi }$ gives $e^{\pi }\approx 8\pi -2.$ Rewriting the approximation for $e^{\pi }$ and using the approximation for $7\pi \approx 22$ gives $e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.$ Thus, rearranging terms gives $e^{\pi }-\pi \approx 20.$ Ironically, the crude approximation for $7\pi$ yields an additional order of magnitude of precision.
Another example involving these constants is: $e+\pi +e\pi +e^{\pi }+\pi ^{e}=59.9994590558\ldots$