2.2 million?

[Prime95]

Are you now going to start on exponents above 2.2 million to try and capture the largest non-Mersenne prime title?

[wfgarnett3]

Hi George,

Nah, we are going to stick where we are. No reason to waste all the work we have done, as we are practically done sieving and have a good amount of PRP work done. So we will stick with searching for the largest "proth" (now 6th largest prime overall). After all, the larger the exponent, the more work that needs to be done; also who knows, someone else may beat Michael Angel number and we would have to start over again :)

regards,
william

[ebx]

A decent CPU can test about 3 131072 gfn numbers every day. Soon we will be out of 'testable' gfn -- all untested numbers are so large that essentially they are in the same scale as mersenne numbers.

Has anyone else thought about LLR, to find primes of the form k*2^n-1?

Since Mersennes primes are a subset of these Riesel primes, there should be plenty of testable k left, that yield frequent primes for n=1,2,3,4....

For example k=195 yields many primes.
I found 195*2^243999-1(73,545 digits)
is prime with LLR, within just a few hours.
In fact there should be k, that for all n are prime, with (k =< 2^n)
I dont expect anyone to find one soon but the implications are fascinating.
Especially with the definition of a Riesel number. A proof may be the only tangible evidence.



PS I notice k is seldomly prime, for Riesel primes.
Does anyone know the mechanism for this?

[cperciva]

Originally Posted by TTn

PS I notice k is seldomly prime, for Riesel primes.
Does anyone know the mechanism for this?

If p divides k, then p does not divide k*2^n-1. This means that N=k*2^n-1 is prime with probability k/phi(k) * 1/ln(N) instead of probability 1/ln(N).

For k<100 with k*2^n-1 prime, this moves the probability that k is prime from 1 in 4 to 1 in 7; for k<1000, the probability of k prime moves from 1 in 6 to 1 in 11; for k<10000, the probability moves from 1 in 8 to 1 in 16.