Re: More magic: Exp(Pi*Sqrt(n))

[Marcus Hutter]

Hi Saibal,   Very interesting! I didn't know that one. The chances that this is just a coincidence seems very low, but   1) it should be verified whether this property holds to much larger numbers, which should be no problem with nowadays computers,   2) Whether it holds to other bases, e.g. binary   If (1) AND (2) are true I imagine a solution to the puzzle.   If (1) OR (2) are false, it may enqueue in the list of unsolved problems for a long time.   Best wishes,   Marcus   P.S. My favority unsolved math problem is Ulam's sequence.   -------------------------------- Dr. Marcus Hutter, IDSIA Istituto Dalle Molle di Studi sull'Intelligenza Artificiale Galleria 2 CH-6928 Manno(Lugano) - Switzerland Phone:   +41-91-6108668 Fax:  +41-91-6108661 E-mail [EMAIL PROTECTED]   http://www.idsia.ch/~marcus Kolmogorov complexity mailing list. at http://www.idsia.ch/~marcus/kolmo.htm ----- Original Message ----- From: Saibal Mitra To: FoR Cc: everything Sent: Friday, August 09, 2002 5:56 PM Subject: More magic: Exp(Pi*Sqrt(n)) Exp(Pi*Sqrt(n)) Page This table lists values of Exp(Pi*Sqrt(n)), for some selected values of n up to 1000. Some of these values are very close to integers. A prize will be awarded to anyone who can either convincingly argue that this is coincidence, or who can explain why this is so in terms intelligible to an intelligent college senior. Something else that might help to lift the veil on this mystery would be a predictor: for a given value of n, is Exp(Pi*Sqrt(n)) close to an integer? -1 -1.0000000000000 0 1.0000000000000 6 2197.9908695437080 17 422150.9976756804516 18 614551.9928856196354 22 2508951.9982574244671 25 6635623.9993411342332 37 199148647.9999780465518 43 884736743.9997774660349 58 24591257751.9999998222132 59 30197683486.9931822609282 67 147197952743.9999986624542 74 545518122089.9991746788535 103 70292286279654.0019412888758 148 39660184000219160.0009666743585 149 45116546012289599.9918302870003 163 262537412640768743.9999999999992 164 296853791705948489.0026726248354 177 1418556986635586485.9961793552497 205 34268610654606782799.0030258870981 223 236855705574162154847.0034451037730 226 324394960614997599147.0065272185438 232 604729957825300084759.9999921715268 267 19683091854079461001445.9927370407698 268 21667237292024856735768.0002920388424 326 4309793301730386363005719.9960116516268 359 70997279226412702087506048.0094309359706 386 639355180631208421212174016.9976698325078 522 14871070263238043663567627879007.9998487264827 566 288099755064053264917867975825573.9938983115610 630 17602513749954237250474851101885772.0095551338274 638 28994858898043231996779771804797161.9923729395451 652 68925893036109279891085639286943768.0000000001637 719 3842614373539548891490294277805829192.9999872495660 790 223070667213077889794379623183838336437.9920551177281 792 249433117287892229255125388685911710805.9960973230079 928 365698321891389219219142531076638716362775.9982597470174 940 677621063891416076248230276783145121158916.0018892548309 986 6954830200814801770418837940281460320666108.9946496112506