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
