There is a theorem by Billingsley which says that if p_i are the prime factors of n in decreasing order, then (log p_i/log n) coverges to samples from a Poisson process with intensity 1 as n goes to infinity. Victor On Sun, Apr 30, 2017 at 02:50 Dan Asimov <dasimov@earthlink.net> wrote:
This is a super interesting question.
Part of the problem is that
IF the distribution* of prime factors DOES approach a limit:
THEN *in what sense* does it approach the limit?
Because it certainly isn't the standard limit.
Maybe in a lim inf or lim sup sort of a way?
Probability theory has a sense of convergence for continuous distributions on the reals that as I recall is, you compare two distributions by taking the difference of their *cumulative* distribution functions, and integrate something like the square of that difference over R.
Maybe something like this works, I don't know.
—Dan
On Apr 29, 2017, at 3:25 PM, rcs@xmission.com wrote:
what a typical factorization looked like for large integers
—Dan ————————————————————————————————————————————— * In *some* sense of the word "distribution". _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun