A strange new (?) fact about Mersenne factors

[Zeta-Flux]

Alex,

I knew I had read a really good paper on Artin's conjecture. I just found a copy. Go to: http://www.mast.queensu.ca/~murty/index2.html

Click on #31.

Enjoy.

-Pace

[akruppa]

Pace,

thanks for the pointers! I'll look into the papers.

Alex

[ChriS]

I returned to the original answer of my problem and thought, this might give a hint for a criterion whether q divides Mp:

Originally Posted by akruppa

This is well known and easily shown with some elementary group theory.

Let p, q be prime, q≡1 (mod p) and q≡±1 (mod 8), 2k=(q-1)/p.
q|Mp iff ord_q(2)=p. The subgroup of elements of order p in Z/Zq has p-1 elements.

...

If ord_q(2) = p, 2 can't be a primitve root for q, at least if q|Mp. Now I checked several values k and 2 <= p < 1000000, and I found that under the preconditions given here (p prime, q = 2kp+1 prime, q = +/-1 mod 8), 2 never seems to be a primitve root for q, regardless whether q|Mp.

Is there an explanation for this? Doesn't that contradict the randomness of primitive roots?

[akruppa]

The condition q≡±1 (mod 8) establishes that 2 is a quadratic residue (mod q), hence it can't be a primitive root. See Gauss' Quadratic Reciprocity Theorem.

Alex