Bill Gosper:
Has anyone listed the largest terms in Hans's data?
Hans Havermann:
I have. :)
For his 13th birthday, funster Neil Bickford has just computed and verified 485M partial quotients (http://www.neilbickford.com/picf.htm). The longest previous CF we can find remains Hans Havermann's 180M, wherein Neil found two glitches--a "q" instead of a "1" at position 4,311,037 (a picked 2^6 bit) and a semicolon following term 100,000,000. The probably biggest news is the survival of 878783625 as the largest known partial quotient, seeming to signal the end of a pattern of "premature" high water marks, assuming the lg 1+t distribution for the tails of almost all reals. If pi is to maintain its reputation, the next high should be a doozy. (I dimly recall deriving that lg 1+t implies that, in an n term burst, the largest should be about e*n. Can anyone help here?) Neil also reports "Taking 9,701 seconds (161 minutes), the longest string of second difference zeroes is of length 18, starting at 173,533,907 and ending at 173,533,924." I.e., he has significantly increased the numerators and denominators necessary for any rational linear relationship of pi to e. Expect details shortly at http://nbickford.wordpress.com/ . Is there reason to expect the longest burst of vanishing second differences should have a difference interval of 1? Or did Neil neglect to mention that it was >1? --rwg