# 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.)[1]

In recreational mathematics an almost integer is any number that is very close to an integer. Well-known examples of almost integers are high powers of the golden ratio $\phi=\frac{1+\sqrt5}{2}\approx 1.618\,$, for example:

• $\phi^{17}=\frac{3571+1597\sqrt5}{2}\approx 3571.00028\,$
• $\phi^{18}=2889+1292\sqrt5 \approx 5777.999827\,$
• $\phi^{19}=\frac{9349+4181\sqrt5}{2}\approx 9349.000107\,$

The fact that these powers approach integers is non-coincidental, which is trivially seen because the golden ratio is a Pisot-Vijayaraghavan number.

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:[2]

$e^{\pi\sqrt{43}}=12^3(9^2-1)^3+744-2.225\cdots\times 10^{-4}\,$
$e^{\pi\sqrt{67}}=12^3(21^2-1)^3+744-1.337\cdots\times 10^{-6}\,$
$e^{\pi\sqrt{163}}=12^3(231^2-1)^3+744-7.499\cdots\times 10^{-13}\,$

where

$21=3\times7, 231=3\times7\times11, 744=24\times 31\,$

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 pi and e have often puzzled mathematicians. An example is: $e^{\pi}-\pi=19.999099979189\cdots\,$ To date, no explanation has been given for why Gelfond's constant ( $e^{\pi}\,$ ) is nearly identical to $\pi+20\,$,[1] which is therefore considered a mathematical coincidence.