I rearranged a little to show the survival rate of exponents

*p* depending on the number of factors of

*p*-1:

Code:

p-1 factors #p #survivors #suvival rate
-----------------------------------------------------
2 factors 23895 8278 0.346
3 factors 82261 29439 0.358
4 factors 129215 47299 0.366
5 factors 128489 48040 0.374
6 factors 96016 36367 0.379
7 factors 59951 22935 0.383
8 factors 33171 12596 0.380
9 factors 17458 6684 0.383
10 factors 8659 3265 0.377
11 factors 4257 1686 0.396
12 factors 2064 784 0.380
13 factors 981 369 0.376
14 factors 434 169 0.389

So it looks like those

*p* with few factors in

*p*-1 do, in fact, have a lower chance of surviving trial division. ATH, I assume the number of factors is the number of prime factors with multiplicity in

*p*-1? It might be interesting to make such a table for the number of proper divisors of

*p*-1 as well.

My hypothesis isn't very convincing, though. By the same argument, 2

^{[I]p[/I]}-4 and 2

^{[I]p[/I]}-1 have at most the factor 3 in common, so the number of divisors in

*p*-2 (and

*p*-3 and

*p*-4 etc.) should also affect the probability that Mp is prime.

Alex