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
Your sequence appears in another series sum (but with non-alternating signs) in a 2003 paper (He and Jejjala, "Modular Matrix Models", arXiv:hep-th/0307293v2 , page 20 ) Possibly also related is the discussion at: mathoverflow.net/questions/4775/why-are-powers-of-exppisqrt163-almost-integers(near the end: David Speyer, Kevin Buzzard, and "FC") which appears to use the same power series. Discussions on the seqfan list seem to indicate that conjecture sequences are all right to add. Naturally you'd add references and the hope is that someone else eventually searches for it and helps fill in the missing pieces. - Robert On Tue, Dec 21, 2010 at 21:38, Bill Gosper <billgosper@gmail.com> wrote:
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
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com
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
participants (3)
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Adam P. Goucher -
Bill Gosper -
Robert Munafo