On Tue, Mar 19, 2013 at 6:21 AM, Bill Gosper <billgosper@gmail.com> wrote:
I wonder if this explains a recent frustration. While puzzling out the condition for arithmetic bursts in A129935, I lost faith in 9.0.1 and 8.0.4 numerics and signed up for an Alpha Pro trial. But I couldn't perform the Floor[...] != Ceiling[...] test on numbers as small as 10^10000. --rwg
I believe the WolframAlpha team merely removed the "Power of 10 representation" software module from the repertoire of modules available to the Wolfram Alpha query engine. Here is WolframAlpha output from about a month ago: mrob.com/pub/math/images/20130227-WA-factorials.png Note the two answers which are different: the first answer is 20000! and appears in the same format given by Mathematica, while the second is Gamma[20000] = 19999! and appears in a "10^10^x" format. I did a lot of large numbers calculations and concluded that the "Power of 10 representation" was being computed by an entirely different software package. In particular, they were doing three different methods of approximating the factorial depending on the hugeness of the argument. In addition to Gamma[x-1], there was a range where you'd get x^x (which is too big) and a middle range where they apparently used a truncated non-Stirling series; all were noticeably wrong. After the recent removal of "Power of 10" results from Wolfram Alpha, what remains are the Mathematica-style results, which have always been true in my tests. The magnitude limit on these results, which I said was "about 10^10^9", happens to be "sort of" close to the magnitude limit for floating-point numbers in Mathematica, given by "MaxNumber" and typically either 2^(2^30-128) ~ 1.2334*10^323228458 or 2^(2^31-352) ~ 1.9202*10^646456887: http://mathematica.stackexchange.com/questions/501/what-determines-the-value... The MaxNumber limit is effectively a limit on the size of the "exponent" in the internal binary representation. The limit of the number of digits in the "mantissa" may be, and probably is, quite different. It would take so much more memory and computing time to compute all the digits of 20000!, just to be able to report that it is "1.819206320230345... x 10 ^ 77337", that I highly doubt they're doing that. Or in Mathematica jargon, I think they have WorkingPrecision set to something like 25, unless or until you explicitly request more digits. Somewhat related are SetPrecision and SetAccuracy, which do more specific things but I suppose they have these set to some standard default as well. Which is a long way of saying I think your 10^10000 issues are a matter of BigFloat mantissa precision, and thus unrelated. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com