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Old 2009-06-13, 20:04   #2
akruppa
 
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"Nancy"
Aug 2002
Alexandria

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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
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