Almost integer

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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.

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