----- Original Message ---- From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, December 11, 2007 11:52:05 PM Subject: [math-fun] Least prime divisor probabilities (or, The Lurch of the SubGenius) ... So: suppose each prime 2,3,5,7,...,p_k < sqrt(n) does not divide n while there remain primes p_r in the range p_k < p_r <= sqrt(n). The greater such k, the greater the chance that such p_r | n. But also the greater such k, the greater is the chance that n is prime. How do these two possibilities relate to each other? (I.e., what should I expect, or bet on, as k increases -- that n is prime, or that some larger test prime <= sqrt(n) divides n ???) (For starters, let p_r denote the largest prime <= sqrt(n).) ... I would expect divisibility by different primes to be independent, i.e. knowing that N is divisible by p, or that N is not divisible by p, should not change the probability (1/q) that N is divisible by a different prime q. Each time a new prime is found to not divide N, the probability that N is prime increases. Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ