Dirty Primes

[Cyclamen Persicum]

(10^p)-1 is never prime

You are right...

But we are talking about primes (10^p-1)/9, i.e. about numbers which
have only 111111... in their decimal representation.

It is 2 units( i.e. eleven - 11), 19 units, 23, 317 and so on. Note that reqular mersennes also have only units, but in binary numeration.

[R.D. Silverman]

Originally Posted by Cyclamen Persicum

[i]


<snip>

And indeed the word "Generalized" should mean "the base other than 2".
Generalized Mersenne Primes have a form (b^n-1)/(b-1), n must be prime. It is interesting that the prime page http://primes.utm.edu contains no information about GMP!!! At least, I couldn't find any...
And what about decimal numeration record, i.e. b=10?
I easily found that n=2, 19, 23, 317, 1031...
Please, Robert, give me a reference.
How one can prove the primality of GMP?

N = (10^n-1)/9 is a PRP for n=49081 and 86453 as well.
For prime proof one must use a general prime prover, or find sufficiently
many factors of N-1, and N+1; there is no equivalent LL test.

See the Cunningham book for other bases.