Quote:
Originally Posted by henryzz
What is the record for a non +1 twin prime? Is there a type non +1 that is easy to prove and sieve?

The
top 20 twin primes list consists solely of k*b^n+1 (all but one have b=2). I don't know of any non +1 that are easy to prove, because AFAIK efficient primality proving algorithms are generally based on N+1 or N1 being factored to 33.33% (or PFGW's 'combined' test that uses largelybutnotquite33% factored N+1 and N1, and of course things like Primo/ECPP that can work on any, but I wouldn't call ECPP 'efficient' in this context), so for twin primes, choosing (smooth number)+1 is a clear choice. I don't know if a record list exists for non +1 twin prime, or (probably the same) a pair of twin primes/PRPs, where at least one is only known to be a PRP. Given the obscurity of that, I'd guess my 5789*2^15513+1, +3 pair have a good chance of being that.
Status:
PRPing: n=513365, 209 candidates remain, sieving more before continuing.
proving PRPs: n=4414, 11 candidates remain. (over 1 hour per test; I'm not sure I'll go all the way with this, 2^15513 is starting to look pretty large...might take a few days just for that test)