The strange case of the unentropic entropy and other bedtime stories of primes and people.

As per considerations discussed prior (principally having to do with how the part of the world made up by you sucks irredeemably, by the way) Phuctor now very conveniently links broken keys straight off its stats page. Let's quickly inventory the 36 factorizable moduli therein contained at the present time, and in the process learn a little of that most difficult of mental exercises, statistics :

1. 231 = 3 * 7 * 11
2. 21 = 3 * 7
3. 9 = 3 * 3
4. 17742509903907 = M * 3 * 3 * 3 * 3 * 3 * 17
5. 4294967297 = Mⁱ
6. 73014444049 = M * 17
7. 270582939711 = M * 3 * 3 * 7
8. 4294967297 = M
9. 98784247831 = M * 23
10. 30064771079 = M * 7
11. 12884901891 = M * 3
12. 21474836485 = M * %
13. 4294967297 = M
14. 4294967297 = M
15. 4294967297 = M
16. 12884901891 = M * 3
17. 115964117019 = M * 3 * 3 * 3
18. 12884901891 = M * 3
19. 38654705673 = M * 3 * 3
20. 64424509455 = M * 3 * 5
21. 12884901891 = M * 3
22. 4294967297 = M
23. 46707769354875 = M * 3 * 5 * 5 * 5 * 29
24. 29 = 29
25. 433791696997 = M * 101
26. 4294967297 = M
27. 29 = 29
28. 83 = 83
29. 64424509455 = M * 3 * 5
30. 115964117019 = M * 3 * 3 * 3
31. 4294967297 = M
32. 20302310412919 = M * 29 * 163
33. 30064771079 = M * 7
34. 12884901891 = M * 3
35. 4294967297 = M
36. 133143986207 = M * 31

We notice at this point that the 36 moduli in question can be readily classified in two sets : those with M as a factor and those without. We will call these sets M+ and M- for convenience.

Set M+ contains 30 elements, of which 21 contain other factors besides M and 9 are equal to M. Out of the prime numbers smaller than or equal to 31 there appear among the factors other than M : 3, 23 times ; 5, 5 times ; 7, 3 times ; 11, 0 times ; 13, 0 times ; 17, 2 times ; 19, 0 times ; 23, 1 time ; 29, 2 times ; 31, one time.

If each number in M+ were random, the expectation would have been for 2 to appear as a factor a total of 15 + 7 + 3 + 1 = 26 times. If each number in M+ were random and odd, the naive expectation would have beenⁱⁱ :

• For 3 to appear a number of 10 + 3 + 1 = 14 times. Instead it appears 23 times.
• For 5 to appear a number of 6 + 1 = 7 times. Instead it appears 5 times.
• For 7 to appear a number of 4 times. Instead it appears 3 times.
• For 11 to appear a number of 3 times. Instead it appears 0 times.
• For 13 to appear a number of 2 times. Instead it appears 0 times.
• For 17 to appear a number of 2 times. It does appear 2 times.
• For 19 to appear a number of 1 time. Instead it appears 0 times.
• For 23 to appear a number of 1 time. It does appear 1 time.
• For 29 to appear a number of 1 time. Instead it appears 2 times.
• For 31 to appear a number of 1 time. It does appear 1 time.

Fortunately for us, the means to evaluate the likeliness of getting in practice a result with a specified probabilityⁱⁱⁱ of occuring has been known for centuries^iv. In our case we can calculate that getting a 1 in 2 event to happen 24 times out of 36 tries exactly is 0.018214307784, the probability to get 24 or more 0.032622667612 and the probability to get 24 or less 0.985591640172.

At this point we could sit and wonder idly at the ~1.8% chance for 3 to appear exactly 24 times!!!(11!1), but this is pretty fallacious : while the odds of any particular combination may be low, it is in fact a given that some sort of combination was going to show up. Consequently, wondering as to why exactly this one came about (as opposed to any other!) is the sort of paranoidly broken mental process that leads one to believe in "higher powers" that "intelligently designed" convenient coincidences and assorted happenstances.

Instead, we will just notice that the probability of the factor count being this spread out from the normal (±6 or more) would be 0.242984954, and move on to the rest. Specifically :

1. The probability of 3 appearing as a factor either more times than 18 + 4 or fewer times than 18 - 4 is 0.242984954.
2. The probability of 5 appearing as a factor either more times than 9 + 4 or fewer times than 9 - 4 is 0.175673693.
3. The probability of 7 appearing as a factor either more times than 6 + 3 or fewer times than 6 - 3 is 0.260920547.
4. The probability of 11 appearing as a factor either more times than 3 + 3 or fewer times than 3 - 3 is 0.167925701.
5. The probability of 13 appearing as a factor either more times than 3 + 3 or fewer times than 3 - 3 is 0.118683859.
6. The probability of 17 appearing as a factor either more times than 2 + 0 or fewer times than 2 - 0 is 0.725764306.
7. The probability of 19 appearing as a factor either more times than 2 + 2 or fewer times than 2 - 2 is 0.265343775.
8. The probability of 23 appearing as a factor either more times than 1 + 0 or fewer times than 1 - 0 is 0.678810524.
9. The probability of 29 appearing as a factor either more times than 1 + 1 or fewer times than 1 - 1 is 0.639964672.
10. The probability of 31 appearing as a factor either more times than 1 + 0 or fewer times than 1 - 0 is 0.63367441.

These are, of course, independent events, the likeliness of n being a divisor is not influenced by and does not influence the likeliness of k being a divisor^v. Consequently, the likeliness of the whole observed pile of coincidences can be simply computed as a normalized product :

% = (0.242984954 / 0.5) * (0.175673693 / 0.5) * (0.260920547 / 0.5) * (0.167925701 / 0.5) * (0.118683859 / 0.5) * (0.725764306 / 0.5) * (0.265343775 / 0.5) * (0.678810524 / 0.5) * (0.639964672 / 0.5) * (0.63367441 / 0.5)

If we do the math, it comes out that the normalcy of this particular distribution is 1.2049761%, which is to say : if you created 36 random large numbers, and then looked at the small primes that divided them, and repeated the process one thousand times, you would get results this divergent from reasonable, probabilistic expectations about a dozen times total.

Set M- contains a further 6 elements. It is too small to be statistically relevant, but a cursory look suggests if it weren't it'd altogether be more likely to push the % above down rather than back towards the mean.

PS. As is traditional with me, I probably fucked up the math. Point out any errors tyvm.

1. 4294967297 (=6700417 * 641 so not a prime) is obviously a special case, being 100000000000000000000000000000001 in binary (already exhaustively discussed) and so we'll leave it aside (it's been replaced with M throughout the factors list to ease reading). [↩]
2. All values are rounded. The idea is that since all numbers are equally likely to show up (as per the definition of "random", every third one on average should be divisible by 3, and every ninth should be divisible by 9, and so on. [↩]
3. Amusingly, the number of times 2 would appear as a factor in a set of random numbers tends towards the cardinal of the set. The number of times 3 would appear as a factor in the same tends towards half the cardinal of the set. Generally, the number of times n would appear as a factor in a set of random numbers tends towards n-1 * the cardinal of the set.

   Proving this is something you might have fun with! [↩]

4. See Mark Nelson for a very accessible explanation of how this works. [↩]
5. Ahh, you virginal flower of all cryptography, the notion of numbers unrelated in practice! [↩]