# 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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