PRP-1

[preda]

Originally Posted by preda

for p == 1583 423452 213582 178911 805893 942695 192421,
I imagine z(p) does not divide 1620,
yet p divides 2^1620 - 1

I take that back. For that p,
znorder(Mod(2, p)) == 1620

So, using z(p) = znorder(Mod(2, p)),

I think that: p | (2^k - 1) IFF z(p) | k.

Then my question is answered like this: any prime p with z(p) > N does not divide 2^k - 1 for any k in [0..N]. So we translated the question into the distribution of z(p).

[axn]

Originally Posted by preda

I think that: p | (2^k - 1) IFF z(p) | k.

Yep, because that is the very definition of "order".

For your other question, you are looking for the smallest prime p > N+1 with order (p-1). And you should find one within a few tries.

[preda]

For a prime p, let's call z(p) the "multiplicative order of 2 modulo p" (AKA znorder(Mod(2,p)) in pari-gp).

Experimentally, it seems that the ratio of primes that produce "non-maximal order" (i.e. z(p) < p-1) gravitates towards the Golomb-Dickman constant, http://mathworld.wolfram.com/Golomb-...nConstant.html . Is that so, or just a coincidence?

(z(p)<p-1 means z(p)<=(p-1)/2)

[preda]

Some information about the implementation in GpuOwl can be found in the GpuOwl thread:
https://mersenneforum.org/showthread...607#post498607

[preda]

Originally Posted by preda

For a prime p, let's call z(p) the "multiplicative order of 2 modulo p" (AKA znorder(Mod(2,p)) in pari-gp).

Experimentally, it seems that the ratio of primes that produce "non-maximal order" (i.e. z(p) < p-1) gravitates towards the Golomb-Dickman constant, http://mathworld.wolfram.com/Golomb-...nConstant.html . Is that so, or just a coincidence?

(z(p)<p-1 means z(p)<=(p-1)/2)

it seems that the ratio of primes for which 2 is a primitive root is given by Artin's constant http://mathworld.wolfram.com/ArtinsConstant.html

interesting that Artin's constant is very close to (1 - Golomb-Dickman constant)

[Dr Sardonicus]

Originally Posted by preda

it seems that the ratio of primes for which 2 is a primitive root is given by Artin's constant http://mathworld.wolfram.com/ArtinsConstant.html

interesting that Artin's constant is very close to (1 - Golomb-Dickman constant)

Coincidentally, I recently compiled some related empirical data, which I posted here.