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  • $\begingroup$ That's a new numerology question, see mathoverflow.net/questions/28088 and especially Gjergji's comment and my answer. I'll probably should ask why $e^\pi-\pi$ is close to an integer. :-) $\endgroup$
    – Wadim Zudilin
    Commented Jul 6, 2010 at 23:14
  • $\begingroup$ Exp[PiSqrt[22]]=2508951.998, Exp[PiSqrt[37]]=199148647.99998, Exp[Pi*Sqrt[58]]=24591257751.9999998. Do you still think this is a coincidence?! $\endgroup$
    – Steven Heston
    Commented Jul 7, 2010 at 2:31
  • 2
    $\begingroup$ @Wadim. I think there is a question here, although no doubt there's a standard answer (but I don't know it; I would start by looking in Cox' book). The question is this. If $x=j(sqrt(-58))$ then $x$ is a root of a monic quadratic with integer coefficients. That quadratic is $x^2 - 604729957849891344000x + 14871070713157137145512000000000$. Furthermore, $e^{\pi\sqrt{58}}+744$ is within $10^{-5}$ of one of the roots. That much isn't surprising at all. What is a little surprising, to me, is that both roots of the quadratic are within $10^{-5}$ of integers. $\endgroup$
    – Kevin Buzzard
    Commented Jul 7, 2010 at 6:08