I just uploaded the file gPEAC.2-131540.txt.gz which contains the results of a long scanning campaign in early February.

From 2 to 131540, there are 13646 maximal and/or perfect sizes, among them 3858 are perfect.

I should plot the density vs value, and the gaps, or histograms thereof...

One very interesting and unexpected find (already noted at wow.) is that :

• Maximal (semi-perfect) systems have a modulus ending with the decimal digits 0, 1, 3, 4, 5, 6, or 9.
• Perfect systems have a modulus ending with the decimal digits 0, 4, 6, or 8.

Weren't we looking for some sort of pattern ? Well, there could be one and I included it in the latest scanner's code: scanning stops if the last digit is either 2 or 7 (with the exceptions of size=2).

There is a blind spot however: the scan only cared for 2 cases but we would need the whole data, including the length and count of all the orbits, in order to "predict" if a given modulus is promising (at least we can eliminate some candidates.

Let's review the actual moduli of the binary PEAC

 1 2 perfect (a noted outlier and exception)
 2 4 maximal
 3 8 maximal
 4 16 maximal
 5 32
 6 64
 7 128
 8 256
 9 512 (impossible)
10 1024 maximal
11 2049
12 4096
13 8192 (impossible)
14 16384
15 32768
16 65536 maximal
17 131072 (impossible)
18 262144
19 524288
20 1048576
21 2097152 (impossible)
22 4194308
23 8388608
24 16777216
25 33554432 (impossible)
26 67108864 perfect ?

From there we can know which powers of 2 to skip:

27 134217728
28 268435456
29 536870912 (impossible)
30 1073741824
31 2147483648
32 4294967296
33 8589934592 (impossible)
34 17179869184
etc.

Better : this can help with finding better candidates for the hash moduli.

sqrt(2^33) = 92681
sqrt(2^35) = 185363
sqrt(2^37) = 370727 (impossible)
sqrt(2^39) = 741455
sqrt(2^41) = 1482910 maybe perfect
sqrt(2^43) = 2965820 maybe perfect
sqrt(2^45) = 5931641
sqrt(2^47) = 11863283
sqrt(2^49) = 23726566 maybe perfect
sqrt(2^51) = 47453132 (impossible)
sqrt(2^53) = 94906265
sqrt(2^55) = 189812531
sqrt(2^57) = 379625062 (impossible)
sqrt(2^59) = 759250124 maybe perfect
sqrt(2^61) = 1518500249
sqrt(2^63) = 3037000499
sqrt(2^65) = 6074000999
sqrt(2^67) = 12148001999
sqrt(2^69) = 24296003999
sqrt(2^71) = 48592007999
sqrt(2^73) = 97184015999
sqrt(2^75) = 194368031998 maybe perfect
sqrt(2^77) = 388736063996 maybe perfect
sqrt(2^79) = 777472127993
sqrt(2^81) = 1554944255987 (impossible)
etc.

But this is for now purely observational and 2^3=8 is an exception to the semi-perfect rule. I'm sure I miss something.

So not only should I recompute these numbers completely, but also plot them to BMP and count the number of orbits, like I did for the videos.

I should also examine the existing list with other bases (not just base 10, but prime bases, so maybe there is another link with Galois Fields ?) and understand why 2 and 7 are exceptions (note: they are 5 apart, mod 10)

This is really so strange... I suspect some prime-number tricky business here.