# Mersenne Digest V1 #1094

Mersenne Digest       Tuesday, December 2 2003       Volume 01 : Number 1094

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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.

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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
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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
_________________________________________________________________________
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_________________________________________________________________________
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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

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End of Mersenne Digest V1 #1094
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