Mersenne Digest Tuesday, December 2 2003 Volume 01 : Number 1094
---------------------------------------------------------------------- Date: Sat, 22 Nov 2003 21:48:19 +0000 From: danny clapp <[EMAIL PROTECTED]> Subject: Mersenne: Re: 40th Mersenne Prime found Great, Well done to all involved. This is a great historic moment for GIMPS. I have not been too involved in the search as of yet, although I will now try to make some more time and join in actually searching. Although I have been on the list and reading all digests for a long time. Again well done, Congratulations. Danny. ___________________________________________________ Cost effective technology solutions for business. Sign up for a free trial today! http://www.officemaster.net _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Sun, 23 Nov 2003 08:53:55 +0000 From: "Brian J. Beesley" <[EMAIL PROTECTED]> Subject: Mersenne: Generalized Mersenne Numbers Congratulations on the (unverified) discovery of the 40th Mersenne Prime. I was thinking (always dangerous!) about generalizing Mersenne numbers. The obvious generalization a^n-1 is uninteresting because they're all composite whenever a>2 and n>1. However there is an interesting generalization: Define GM(a,b) = a^b-(a-1), so GM(2,b) = M(b); also GM(a,1) = 1 for all a The distribution of primes amongst GM(a,b) for small a > 2 and small b does seem to be interesting - some values of a seem to yield a "richer" sequence of primes than others. Note also that, in this generalization, some _composite_ exponents can yield primes. Another interesting point: the "generalized Mersenne numbers" seem to be relatively rich in numbers with a square in their factorizations - whereas Mersenne numbers proper are thought to be square free. (Or is that just Mersenne numbers with prime exponents?) A few interesting questions: (a) Is there a table of status of "generalized Mersenne numbers" anywhere? (b) Is there a method of devising Lucas sequences which could be used to test GM(a,b) for primality reasonably efficiently? (c) Are there any values of a which result in all GM(a,b) being composite for b>1? (There are certainly some a which result in the first few terms in the sequence being composite e.g. GM(5,2) = 21, GM(5,3) = 121 & GM(5,4) = 621 are all composite - but GM(5,5) = 3121 is prime). (d) Is there any sort of argument (handwaving will do at this stage) which suggests whether or not the number of primes in the sequence GM(a,n) (n>1) is finite or infinite when a > 2? Regards Brian Beesley _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Sun, 23 Nov 2003 08:53:55 +0000 From: "Brian J. Beesley" <[EMAIL PROTECTED]> Subject: Mersenne: Generalized Mersenne Numbers Congratulations on the (unverified) discovery of the 40th Mersenne Prime. I was thinking (always dangerous!) about generalizing Mersenne numbers. The obvious generalization a^n-1 is uninteresting because they're all composite whenever a>2 and n>1. However there is an interesting generalization: Define GM(a,b) = a^b-(a-1), so GM(2,b) = M(b); also GM(a,1) = 1 for all a The distribution of primes amongst GM(a,b) for small a > 2 and small b does seem to be interesting - some values of a seem to yield a "richer" sequence of primes than others. Note also that, in this generalization, some _composite_ exponents can yield primes. Another interesting point: the "generalized Mersenne numbers" seem to be relatively rich in numbers with a square in their factorizations - whereas Mersenne numbers proper are thought to be square free. (Or is that just Mersenne numbers with prime exponents?) A few interesting questions: (a) Is there a table of status of "generalized Mersenne numbers" anywhere? (b) Is there a method of devising Lucas sequences which could be used to test GM(a,b) for primality reasonably efficiently? (c) Are there any values of a which result in all GM(a,b) being composite for b>1? (There are certainly some a which result in the first few terms in the sequence being composite e.g. GM(5,2) = 21, GM(5,3) = 121 & GM(5,4) = 621 are all composite - but GM(5,5) = 3121 is prime). (d) Is there any sort of argument (handwaving will do at this stage) which suggests whether or not the number of primes in the sequence GM(a,n) (n>1) is finite or infinite when a > 2? Regards Brian Beesley _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ Date: Tue, 02 Dec 2003 13:01:48 -0500 From: George Woltman <[EMAIL PROTECTED]> Subject: Mersenne: 40th Mersenne Prime verified and word is getting out Hi all, Michael Shafer discovered the 40th known Mersenne prime, 2^20996011-1. Congratulations Michael. This prime is over 6.3 million digits, beating the previous world record prime by over 2 million digits. Scott has handed out the press release and already the first online article of the discovery has appeared: http://www.newscientist.com/news/news.jsp?id=ns99994438 You can also read Scott's press release http://www.mersenne.org/20996011.htm I'll try to add more links to the http://mersenne.org web page as they become available. Well done everyone, George _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers ------------------------------ End of Mersenne Digest V1 #1094 *******************************
