# Re: Mersenne: Generalized Mersenne Numbers

Brian J. Beesley <[EMAIL PROTECTED]> writes
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
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