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?
Some time ago I had a look at numbers of the form 2^n - 3, i.e. GM(4, n/2). Here are my results for 3320 <= n <= 16800:
2^n - 3 is a verified prime for n = 3954, 5630, 6756, 8770, 10572, 14114.
2^n - 3 is a probable prime for n = 14400, 16460, 16680.
I don't know if someone else has verified the last three. Also
2^12819 - 7 (GM(8, 4273)) is a probable prime,
2^8824 - 15 (GM(16, 2206)) is a verified prime.
The verified primes were done by factorization of N+1 and N-1, and APRCL.
-- Tony _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
