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I've heard about Heegner numbers some time ago and have been playing today with their property which seems very intriguing to me (and, to be fair, the one I can understand most easily), namely, the fact that $e^{\pi\sqrt{d}}$ turns out to be quite close to an integer when $d$ is a Heegner number. For example, somewhat most famously, $e^{\sqrt{163} \pi } =262537412640768743+0.9999999999993\ldots$

So, I've just computed the numbers $\exp(\pi d^p)$ for $p$ in $(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5})$ and $d$ from $1$ to $10000$ with this code in Mathematica (if anyone is interested):

x = 3000
DeleteCases[
  Prepend[Map[If[(# > 10^-4) && ((1 - #) > 10^-4), 0, #] &, 
    N[Table[{FractionalPart[N[Exp[\[Pi] Sqrt[i]], x]], 
       FractionalPart[N[Exp[\[Pi] i^(1/3)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(2/3)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(1/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(2/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(3/5)], x]], 
       FractionalPart[N[Exp[\[Pi] i^(4/5)], x]], i}, {i, 10000}], 
     15], {2}], {1/2, 1/3, 2/3, 1/5, 2/5, 3/5, 4/5}], {0, 0, 0, 0, 0, 
   0, 0, _?NumberQ}] // TableForm

There was some list of numbers, from which I found that $e^{\sqrt[3]{3163} \pi } = 106618076286425992470 + 0.99997\ldots\\$, $e^{\sqrt[3]{7577} \pi } = 627840988101335617898564443+0.99991\ldots\\$, $e^{\sqrt[5]{7607} \pi } = 141371353+0.000010\ldots$ and a bunch of other examples (nothing so striking as $e^{\sqrt{163} \pi }$ though).

Is there some reason for this (maybe expansion of some generalization of j-invariant) or is it just a byproduct of taking huge amount of numbers?

user108687
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    Seems to be pure coincidence. The fractional part of those expressions can be expected to behave like a pseudorandom sequence in $[0,1]$, so from time to time we will hit almost-integer values. By analyzing the fractional part of $e^n$, you will get similar results. – Peter Aug 31 '21 at 11:15
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    There are already several posts in this direction, e.g., here, or here, etc. – Dietrich Burde Aug 31 '21 at 11:16
  • FWIW, $e^{5\pi}$ also happens to be an almost-integer. – PM 2Ring Aug 31 '21 at 11:51
  • @DietrichBurde could you elaborate? In these posts there are given reasons behind similar facts, and I don't understand whether you are saying there's a reason behind my examples as well or it's just a coincidence. – user108687 Aug 31 '21 at 12:11
  • It has been elaborated in these posts:"Close" to an integer says not much, because "close" is up to interpretation, it is relative. So Peter's comment says it. Of course, there are sometimes mathematical objects behind this, but not an enough precise statement why this is meaningful for being close to an integer. – Dietrich Burde Aug 31 '21 at 12:18

0 Answers0