Firstly, these almost integers occur for many values of e^(pi*sqrt(n)): http://www.mathematik.uni-bielefeld.de/~sillke/NEWS/exp-sqrt Large Heegner numbers have this property, and so do (obviously) numbers of the form H*m², for small m. Setting H = 163, m = 2 gives the explanation for why e^(pi*sqrt(652)) is an almost integer, being (e^(pi*sqrt(163)))^2. The explanation lies in the q-expansion of the incredible j-function: 1/q + 744 + 196884q + 21493760q² + 864299970q³ + ... The coefficients of the j-function can be determined by the Monster Group. (!) Sincerely, Adam P. Goucher ----- Original Message ----- From: "Bill Gosper" <billgosper@gmail.com> To: <math-fun@mailman.xmission.com> Sent: Wednesday, December 22, 2010 2:38 AM Subject: [math-fun] e^(pi rt 163) =
s^3 + 744 - 196884/s^3 + 167975456/s^6 - 180592706130/s^9 + 217940004309743/s^12 - 19517553165954887/s^15 + 74085136650518742/s^18 - ...
where s = 640320. Only the 1st three terms are in EIS. Are the rest well defined? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun