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Old 2018-12-17, 08:03   #62
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Sep 2003

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Default A skew also seems to occur for fully-factored exponents, but this time for p=3 (mod 4)

There are 325 currently-known exponents of fully-factored or probably-fully-factored Mersenne exponents.

Overall, there is a slight prevalence of p=3 (mod 4). But it seems to get a lot bigger when we consider only larger p.

For all p, here are the numbers, broken down by mod 4 and by mod 8.

mod 4
    149 p=1
    176 p=3

mod 8
     75 p=1
     86 p=3
     74 p=5
     90 p=7
However, this set contains a lot of very small exponents. About 55% of the currently known fully-factored or probably-fully-factored Mersenne numbers are in the Cunningham range of exponents smaller than 1200.

If we consider only exponents larger than 1200, the numbers are:

mod 4
     60 p=1
     83 p=3

mod 8
     34 p=1
     38 p=3
     26 p=5
     45 p=7
If we consider only exponents larger than 10000, the numbers are:

mod 4
     28 p=1
     45 p=3

mod 8
     17 p=1
     21 p=3
     11 p=5
     24 p=7
If we consider only exponents larger than 100000, the numbers are:

mod 4
     11 p=1
     24 p=3

mod 8
      6 p=1
      8 p=3
      5 p=5
     16 p=7
If we consider only exponents larger than 1M, the numbers are:

mod 4
      3 p=1
      9 p=3

mod 8
      2 p=1
      3 p=3
      1 p=5
      6 p=7
Obviously, "fully factored" is a nebulous concept that evolves over time, as we find more factors and as we PRP test the known cofactors. Because PRP cofactor testing was only introduced to Primenet about a year ago, we've only PRP cofactor tested exponents up to about 8.6M so far.

Based on the data in the earlier tables, our factor finding is not biased in favor of p=1 (mod 4) or p=3 (mod 4), other than the small effect of Wagstaff's heuristic. And our PRP testing is similarly not biased, it just advances in a wavefront, testing all exponents in its path.

And yet the skew for fully-factored and probably-fully-factored Mersenne numbers in favor of p=3 (mod 4) only seems to get stronger as we increase the threshold for filtering out small exponents. And the last two sets even show a differentiation in favor of p=7 (mod 8).

The statistics are probably too small to be significant, but it's intriguing...

Last fiddled with by GP2 on 2018-12-17 at 08:58
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