Quote:
Originally Posted by Madpoo
I ran a quick count of the distinct exponents with any known factors and whether that exponent is 1 or 3 mod 4:
Mod Count
1 14,559,812 (49.301%)
3 14,972,752 (50.699%)
I wouldn't say it's a runaway for the 3 mod 4 exponents being factored. In fact it seems like a pretty basic variation around 50/50 based on an incomplete set.

But exponents of p=3 mod 4 do have slightly more factors than exponents of p=1 mod 4, as was
noted as early as 1967 by Ehrman (and cited by Wagstaff in 1983).
So I would expect that the counts above will never be exactly 50/50 even theoretically. If you slightly alter the λ of a Poisson distribution then you slightly alter your counts at k=0.
We could look at the factors themselves, rather than distinct exponents with or without factors. Based on the same dataset I used before (frozen on 20181201), I get:
Code:
all f f <= 65 bits
p=1 mod 4, f=1 mod 8 9846883 9304449
p=3 mod 4, f=1 mod 8 9833742 9310151
p=1 mod 4, f=7 mod 8 10655959 10119410
p=3 mod 4, f=7 mod 8 11565454 11045807
So
 There are more factors with f=7 (mod 8) than f=1 (mod 8)
 The fact that exponents with p=3 (mod 4) are more likely to have factors than exponents with p=1 (mod 4) is due to the factors that are 7 (mod 8).
 As your post also reflected, factors of size 65 bits or less are a pretty big subset of the set of all known factors
If p=3 mod 4 and 2p+1 is prime, then it's a factor of the Mersenne number with exponent p, and f=2p+1=7 mod 8. And if p=1 mod 4 then the smallest possible factor is 6p+1, and again in this case there will be f=6p+1=7 mod 8.