For most players, it's the fun of the hunt. There is a prize of (I think) $100K from EFF for the first 10M digit prime. I don't know if/how it would be divided if a GIMPS player wins. Unless you've got inside information on likely exponents to try, the economics are unfavorable: you would have to spend more than $100K on computers to have a good chance of winning the prize. The consensus conjecture is that there are an infinity of Mersenne primes, with the number up to exponent N being roughly K log N. Various values of K have been offered. The evidence is mainly the empirical data, for what it's worth. It's known that the potential divisors of 2^P-1 (P prime) are restricted to the arithmetic sequences 2kP+1 and 8k+-1. But these divisors seem to have a correspondingly increased likelihood of actually dividing 2^P-1, so the net effect on the chances of 2^P-1 being prime (versus a randomly selected integer of about the same magnitude) seems minor. AFAIK, there's no information to support any guess such as P = 1 (mod 7) are more or less likely to be a winning exponent. The only similar result is that, if P = 4k+3 and Q = 2P+1 is prime, then Q | 2^P-1. Example: P=11, Q=23, 2^11-1 = 23*89. Presumably R = 6P+1 and S = 8P+1 also have good chances of dividing 2^P-1. The condition for R is that P = 4k+1 and that 2 is a cubic residue (mod R). The cubic residue condition has chance 1/3 (proven?), and is equivalent to the unique representation of R = X^2 + 3 Y^2 having 3|Y. Example: P=37, R=223, R = 14^2 + 3 * 3^2, 2 is a cubic residue mod 223, 68^3 = 2 (mod 223), 2^37-1 = 223 * 616318177. Similarly for S. I don't know if Euclid said it explicitly, but the infinitude of perfect numbers and Mersenne primes is an obvious question to ask once he proves the formula 2^(P-1) * (2^P-1) generates perfect numbers. This might be the oldest outstanding math conjecture. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Daniel Asimov Sent: Fri 12/30/2005 11:09 PM To: math-fun Subject: Re: [math-fun] New Mersenne prime, exponent 30,402,457 Rich wrote: << There's a new Mersenne prime, 2 ^ 30,402,457 - 1. 9.2 million digits. More details at www.mersenne.org.
I'm curious if people are searching for new Mersenne primes just for the fun of the hunt, or if there is a deeper reason, such as to gather evidence bearing on some conjecture(s) about them (or evidence for potentially formulating some conjecture about them). Question: Is it known that there exist infinitely many Mersenne primes? If not, what is the evidence for and against? --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun