I'd look into trying that to Crandall's Theorem re: the minimum length of a non-trivial cycle, Steiner's result regarding the (essentially) non-existence of non-trivial cycles with the property of being circuits, as well as Baker-Feldman's result that there exists c > 0 such that for all k, l > 1, | k log 2 - l log 3 | >= k^-c The Wikipedia article states the Khinchin constant property does not apply to rationals. But the results above are tied to the continued fraction of log_2 3. Andres. On 3/24/18 11:55 , Bill Gosper wrote:
There are 356033 integers within 41 Collatz (#/2 | (3#+1)/2) steps of 1. 356033/(4/3)^41 = 2.685357138639965 . . . 843774/(4/3)^44 = 2.684860516180538 . . . 63149973/(4/3)^59 = 2.685268450566904 . . . 112268898/(4/3)^61 = 2.685322969193732
But N@Khinchin = 2.685452001065306
(Thank Julian for the 4/3.) Does anyone recall an objection that the usual definition isn't really the expected geometric mean because its derivation assumes consecutive partial quotients are independent? Is there a "CorrectedKhinchin"? https://en.wikipedia.org/wiki/Khinchin%27s_constant and http://mathworld.wolfram.com/KhinchinsConstant.html say flatly No. (My Finch is on loan.) --rwg (EWW: In (8), the ln k belongs in the exponent.) Happy 8th, Gabriel! _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun .