Quote:
Originally Posted by R.D. Silverman
You really should consider just a SINGLE MR test followed by either a
Lucas test or a FrobeniusGrantham. There is no known pseudoprime
for a single MR followed by a single Lucas test. (Preda Mihailescu
and I looked for one without success).

That's what I already use for daytoday probableprime testing.
Quote:
Originally Posted by R.D. Silverman
If the objective is proved primes, one can just trial divide N1 and N+1
to N^(1/6), and apply the SelfridgeLehmerBrillhart test that allows
one to prove primality if enough factors of N1 and N+1 are known.

Do you think this would be faster than three MR tests for 64bit numbers? Either way you'd start with an MR test to weed out composites, and the combined test (given the factorization) should be slower than a single MR test, so it would be close. Fortunately not much trial division should be needed  though don't you need
factored part at least N^(1/6), not factoring up to N^(1/6)? That matters when you have a prime between two fairly rough numbers (e.g., members of A106639).