A069014 - OEIS

%I #29 Apr 30 2018 11:06:29

%S 1,2,6,17,22,25,37,58,163

%N Integers n at which the difference between e^(Pi*sqrt(n)) and its nearest integer is a new minimum.

%C a(10) > 99999.

%C a(10) > 10^7. - _Jon E. Schoenfield_, Mar 27 2015

%C At n=163, exp(Pi*sqrt(n)) is remarkably close to its nearest integer, differing from it by less than 7.5*10^-13. Consequently, for values of n=163*k^2 where k is a sufficiently small integer greater than 1, exp(Pi*sqrt(n)) = exp(Pi*sqrt(163*k^2)) = exp(Pi*sqrt(163)*k) = (exp(Pi*sqrt(163)))^k will also be close to an integer. If we exclude numbers of the form 163*k^2 for k=2..4, then the number of values of n < 10^7 at which exp(Pi*sqrt(n)) differs from its nearest integer by less than 10^-6 is 21, which is about what we would expect if we were instead generating random numbers whose fractional parts followed a uniform distribution on the interval [0,1). If the fractional parts continue to behave in this way, then we could expect about a 50% chance of finding a(10) at some value below log(2)/(2u) = 4.62*10^11 where u = abs(t - round(t)) and t = exp(Pi*sqrt(163)). - _Jon E. Schoenfield_, Mar 27 2015

%H University of Sheffield, Department of Pure Mathematics, <a href="http://www.shef.ac.uk/~puremath/theorems/nearint.html">Is e^(Pi*Sqrt(163)) an integer?</a>

%H University of Sheffield, Department of Pure Mathematics, <a href="http://web.archive.org/web/20040818223300/www.shef.ac.uk/puremath/theorems/nearint.html">Is e^(Pi*Sqrt(163)) an integer?</a>

%t s = 1; Do[ t = Abs[ N[ E^(Pi*Sqrt[n]), 10^3] - Round[ E^(Pi*Sqrt[n])]]; If[s > t, s = Abs[t]; Print[n]], {n, 1, 10^4}]

%o (PARI) A069014()={default(realprecision, 1000);my(maxx=9999);n=1;minn=1;while (n<maxx,q=abs(exp(Pi*sqrt(n))-round(exp(Pi*sqrt(n))));if(q<minn,minn=q;print1 (n,","));n+=1);} \\ _Bill McEachen_, Mar 15 2015

%Y Cf. A014708.

%K nonn,more

%O 1,2

%A _Robert G. Wilson v_, May 24 2002

%E Name edited by _Jon E. Schoenfield_, Mar 24 2015