Mersenne: Generalized Mersenne Numbers

[Wojciech Florek]

Hi!
Jean Penne has written about 3^n-2. I've played for a while with
5^n-(5-1)^5=5^n-1024. Of course, for n=5k they are composite; since
4=2^2 they are also composite for even n. Moreover, there are some series
of composite numbers, e.g. for n=10,16,22,28,... they are divisible by 7
(it can be proven), and for n=3k d=31 is a divisor (I had no time to
prove it, but it is easy, I think). These rules excludes many of numbers,
so for n1000 I've found only three primes:
5^7-1024=77101, 5^11-1024=48827101 and 634-digit one for n=907 (primality
tested by Marcel Martin's Primo.

Regards 
Wojtek (WsF)

===
Wojciech Florek (WsF)
Adam Mickiewicz University, Faculty of Physics
ul. Umultowska 85, 61-614 Poznan, Poland

phone: (++48-61) 8295033 fax: (++48-61) 8295167
email: [EMAIL PROTECTED] 







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Re: Mersenne: Generalized Mersenne Numbers

[Tony Forbes]

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 a2 and n1. 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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