Some of this is likely to be based on the density of primes. The "Prime Number Theorem" shows that the asymptotic density of primes is x / ln x. This density is often written pi(x) [the lower case Greek letter, btw] with pi(x) = (the number of primes less than or equal to x) / x. This is not a trivial result and it has been over 30 years since I looked at it. There is an "elementary" proof [nothing more than calculus] that takes about 40 pages, as I recall. This spends a lot of time on Mobius inversions -- but the details are gone without going back to sources. [It might be just the number of primes less than x, but for large x this is pretty similar.] AFAIK, the question of the number of Mersenne primes is open. (That is: finite or infinite set.) Joth ----- Original Message ----- From: Walt Mankowski <[EMAIL PROTECTED]> To: mersenne <[EMAIL PROTECTED]> Sent: Tuesday, October 12, 1999 8:33 PM Subject: Re: probability of primeness (was: Re: Mersenne: splitting up 10m digit primes) > On Tue, Oct 12, 1999 at 10:53:18PM -0400, Darxus wrote: > > > > I'm hoping what I have to say in this email might be important. > > > > On Tue, 12 Oct 1999, George Woltman wrote: > > > > > At 04:12 PM 10/12/99 -0400, you wrote: > > > >> >And how is the probability of finding a prime calculated ? > > > >> > > > >> It is roughly how-far-factored-in-bits * 2 / exponent > > > > > > > >Okay.. what's "how-far-factored-in-bits" mean ? > > > > > > I think trial factoring is done to 2^68 for an exponent around 33 million. > > > Thus your chance is 2 * 68 / 33000000. > > > > Okay, so as far as we know, each number is equally likely to be prime, and > > this probability is just based on how much has already been tested ? > > No, they're saying the probability is based on how deeply they've > tried to factor the number before trying the LL test. The more > numbers N you rule out as potential factors of M, the more likely M is > to be prime. > > Also, although there are an infinite number of primes, their density > thins out considerably as they get large because there are so many > more potential factors below them. This applies to primes in general; > I don't know if it applies to Mersenne primes. > _________________________________________________________________ > Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm > Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers > _________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
