2 in one...

From: "wouter meeussen" <wouter.meeussen@pandora.be> 'proving' means 2 LL tests (double check),

Which is of course rather silly, mathematically speaking. No non-zero number squared is zero, so you can't say that the second test rules out any chance of there being an error. IMNSHO Its status is proved already, it's now simply being verified. From: Daniel Asimov <dasimov@earthlink.net>

Rich wrote:

<< There's a new Mersenne prime, 2 ^ 30,402,457 - 1. 9.2 million digits. More details at www.mersenne.org.

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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).

The ones that I've met randomly IRL are doing it for glory. The one-in-a-million chance that their name would be at the top of the table, and more recently that they might get a slice of the EFF prize.

Question: Is it known that there exist infinitely many Mersenne primes? If not, what is the evidence for and against?

There's no evidence against, and there are heuristics for. The cyclotomic nature messes with the Nash weight (relative density of primes in a k*2^n+/-1 sequence with fixed k, Mersennes are k=1), but asymptotically only applies a constant scaling factor to the expected density. There's no finite covering set, so the sequence is expected to have an infinitude of primes. More at http://primepages.org/mersenne/index.html and pages linked therefrom Phil () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani] __________________________________ Yahoo! for Good - Make a difference this year. http://brand.yahoo.com/cybergivingweek2005/