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Old 2018-12-19, 08:18   #80
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Sep 2003

5·11·47 Posts

Originally Posted by Madpoo View Post
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 2018-12-01), I get:

                         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
  1. There are more factors with f=7 (mod 8) than f=1 (mod 8)
  2. 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).
  3. 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.

Last fiddled with by GP2 on 2018-12-19 at 08:29
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