# Almost integer Ed Pegg, Jr. noted that the length d equals ${\frac {1}{2}}{\sqrt {{\frac {1}{30}}(61421-23{\sqrt {5831385}})}}$ that is very close to 7 (7.0000000857 ca.)

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 are considered interesting when they arise in some context in which they are unexpected.

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

Well-known 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 countless almost integers, for instance:

• $\operatorname {Fib} (360)/\operatorname {Fib} (216)\approx 1242282009792667284144565908481.999999999999999999999999999999195$
• $\operatorname {Lucas} (361)/\operatorname {Lucas} (216)\approx 2010054515457065378082322433761.000000000000000000000000000000497$

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

• $a(n)=\operatorname {Fib} (45\times 2^{n})/\operatorname {Fib} (27\times 2^{n})\approx \operatorname {Lucas} (18\times 2^{n})$
• $a(n)=\operatorname {Lucas} (45\times 2^{n}+1)/\operatorname {Lucas} (27\times 2^{n})\approx \operatorname {Lucas} (18\times 2^{n}+1)$

As n increases, the number of consecutive nines or zeros beginning at the tenth 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 being 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: $e^{\pi }-\pi =19.999099979189\ldots$  To date, no explanation has been given for why Gelfond's constant ($e^{\pi }$ ) is nearly identical to $\pi +20$ , which is therefore considered a mathematical coincidence.