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Old 2018-12-14, 02:32   #45
Dr Sardonicus
 
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Feb 2017
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I used the formula

round(exp(Euler)*log(p*log(2))/log(2))

on the first 50 (known) Mersenne prime exponents p, with the following result:

[1, 2, 3, 4, 6, 6, 7, 8, 10, 11, 11, 12, 15, 16, 17, 19, 19, 20, 21, 21, 23, 23, 23, 24, 25, 25, 27, 28, 29, 29, 31, 34, 34, 35, 35, 37, 37, 40, 41, 42, 43, 43, 43, 44, 44, 44, 44, 45, 46, 46]

I then subtracted the ith component of this vector from i. This gives the "excess" or "defect" (compared to the formula) of Mersenne primes up to that exponent.

[0, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, -2, -2, -2, -3, -2, -2, -2, -1, -2, -1, 0, 0, 0, 1, 0, 0, 0, 1, 0, -2, -1, -1, 0, -1, 0, -2, -2, -2, -2, -1, 0, 0, 1, 2, 3, 3, 3, 4]

The components up to the 47th are known to be correct, but it is not known whether there are other Mersenne primes between the 47th Mersenne prime and the 50th known Mersenne prime. If there are, one or more of those last three numbers will be bumped up.

Now that doesn't look too bad, does it?

Oh, wait. The count grows like log(p). An additive discrepancy in the count corresponds (roughly) to a multiplicative factor between the actual and predicted prime exponent values. A discrepancy of d means the actual and predicted exponent values will differ by a multiplicative factor of roughly exp(log(2)*d/exp(Euler)), or about 1.47576^d. I say "roughly" because this can itself be off by a factor (either way) of exp(log(2)/2/exp(Euler)), or 1.2148, due to the estimated count being rounded to an integer value.

Finally, I tried a "predicted" set of the first 50 Mersenne prime exponents. I used the formula

nextprime(exp((i - .5)*log(2)/exp(Euler)),

in which I subtracted 1/2 to get the lowest value for which inversion would "round up" to i. The result was

[2, 3, 5, 7, 11, 13, 19, 29, 41, 59, 89, 127, 191, 277, 409, 607, 907, 1319, 1933, 2851, 4211, 6211, 9173, 13523, 19961, 29453, 43481, 64151, 94687, 139697, 206153, 304217, 448969, 662551, 977761, 1442939, 2129443, 3142547, 4637639, 6844039, 10100173, 14905433, 21996881, 32462159, 47906351, 70698347, 104333939, 153971921, 227225797, 335331067]

The actual 50th known Mersenne prime exponent is only 77232917. So from that point of view, the Mersenne primes are piling up a lot faster than expected.
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Old 2018-12-14, 08:16   #46
GP2
 
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Quote:
Originally Posted by GP2 View Post
However, thanks to user TJAOI and his very systematic "by k" factoring, we can be quite certain that we know all factors of bit-length 65 or less of all Mersenne numbers with prime exponent less than 1 billion. As I posted last year in the "User TJAOI thread", empirically we can see that he does not miss any factors, except for one very minor glitch in 2014 that he very quickly corrected.

So we can use our factor data to check if the actual exact number of factors smaller than B = 2^65 matches what Wagstaff's heuristic predicts, for various ranges of p.
Here's a preliminary data table:

The data reflects a snapshot of factor data from December 1, 2018.

The first column of each line is the range of exponents, in M. Each line represents a range of 10M in extent. Maybe these ranges are too broad, but I wanted the big picture, all the way up to exponents of 1 billion.

The second column is the average number of factors-smaller-than-2^65 per exponent in that range. This is averaged over all exponents in that range, including the ones that have no such factors.

The next column shows how many exponents in that range have zero factors-smaller-than-2^65, or indeed, no known factors at all. This includes Mersenne primes. For example, in the range from 0 to 10M, there are 202974 exponents with no known factors and 31745 exponents that have known factors but none of them are smaller than 2^65, for a total of 234719.

The next column shows how many exponents in that range have exactly one factor-smaller-than-2^65. And so forth.

The cutoff point for the table display is seven such factors, although the long tail nearly always has a few exponents with even more than this. The "..." at the end of each line serves as a reminder of this.

A very quick spot check seems to indicate that these figures follow Poisson distributions. For instance, there are 482825 primes between 990M and 1G (the last line of the table), so a Poisson distribution with λ = 0.731 would have 232445 (k=0), 169917 (k=1), 62104 (k=2), 15132 (k=3), 2765 (k=4), 404 (k=5), 49 (k=6), 5 (k=7), which more or less matches the data.

I think these numbers are correct, but there's always the possibility my script was buggy.

Code:
000 1.038 [(0, 234719), (1, 245459), (2, 126571), (3, 43773), (4, 11266), (5, 2285), (6, 434), (7, 61)] ...
010 0.946 [(0, 234696), (1, 223147), (2, 105645), (3, 33204), (4, 7694), (5, 1404), (6, 204), (7, 32)] ...
020 0.918 [(0, 233722), (1, 216123), (2, 99324), (3, 29906), (4, 6778), (5, 1203), (6, 165), (7, 23)] ...
030 0.899 [(0, 233239), (1, 212189), (2, 94611), (3, 28298), (4, 6159), (5, 1091), (6, 177), (7, 28)] ...
040 0.885 [(0, 233527), (1, 208120), (2, 91992), (3, 26797), (4, 5873), (5, 1010), (6, 147), (7, 14)] ...
050 0.875 [(0, 232763), (1, 205642), (2, 90281), (3, 25647), (4, 5559), (5, 960), (6, 112), (7, 16)] ...
060 0.866 [(0, 232940), (1, 203714), (2, 87756), (3, 25142), (4, 5374), (5, 892), (6, 116), (7, 14)] ...
070 0.858 [(0, 232694), (1, 201704), (2, 86510), (3, 24423), (4, 5032), (5, 844), (6, 101), (7, 10)] ...
080 0.850 [(0, 233048), (1, 200298), (2, 84600), (3, 23720), (4, 4978), (5, 812), (6, 101), (7, 15)] ...
090 0.845 [(0, 233394), (1, 198331), (2, 83423), (3, 23650), (4, 4781), (5, 806), (6, 103), (7, 13)] ...
100 0.839 [(0, 233491), (1, 197332), (2, 82338), (3, 23054), (4, 4748), (5, 787), (6, 85), (7, 19)] ...
110 0.835 [(0, 232425), (1, 196537), (2, 81417), (3, 22502), (4, 4563), (5, 763), (6, 117), (7, 12)] ...
120 0.832 [(0, 232832), (1, 195028), (2, 81092), (3, 22123), (4, 4621), (5, 724), (6, 104), (7, 15)] ...
130 0.829 [(0, 232032), (1, 194479), (2, 80456), (3, 21752), (4, 4459), (5, 706), (6, 110), (7, 18)] ...
140 0.823 [(0, 232887), (1, 193075), (2, 79635), (3, 21507), (4, 4297), (5, 702), (6, 84), (7, 9)] ...
150 0.820 [(0, 232208), (1, 193109), (2, 78567), (3, 21156), (4, 4250), (5, 674), (6, 85), (7, 13)] ...
160 0.817 [(0, 232414), (1, 192361), (2, 77925), (3, 21063), (4, 4096), (5, 673), (6, 82), (7, 10)] ...
170 0.815 [(0, 232521), (1, 191197), (2, 77714), (3, 20980), (4, 4160), (5, 622), (6, 95), (7, 11)] ...
180 0.813 [(0, 232084), (1, 190231), (2, 77237), (3, 20611), (4, 4154), (5, 679), (6, 80), (7, 11)] ...
190 0.811 [(0, 231926), (1, 189643), (2, 76797), (3, 20216), (4, 4106), (5, 686), (6, 82), (7, 7)] ...
200 0.805 [(0, 232847), (1, 189089), (2, 76091), (3, 20045), (4, 3972), (5, 568), (6, 67), (7, 8)] ...
210 0.805 [(0, 232215), (1, 188110), (2, 75819), (3, 20208), (4, 3871), (5, 615), (6, 70), (7, 5)] ...
220 0.801 [(0, 232854), (1, 187649), (2, 75160), (3, 19785), (4, 3890), (5, 618), (6, 68), (7, 6)] ...
230 0.799 [(0, 232666), (1, 187423), (2, 74873), (3, 19444), (4, 3886), (5, 597), (6, 71), (7, 9)] ...
240 0.797 [(0, 232433), (1, 187116), (2, 74387), (3, 19365), (4, 3793), (5, 591), (6, 81), (7, 8)] ...
250 0.797 [(0, 231941), (1, 186757), (2, 74021), (3, 19364), (4, 3759), (5, 598), (6, 90), (7, 10)] ...
260 0.795 [(0, 232152), (1, 185969), (2, 73477), (3, 19468), (4, 3788), (5, 584), (6, 80), (7, 4)] ...
270 0.792 [(0, 232465), (1, 185617), (2, 73191), (3, 19002), (4, 3748), (5, 575), (6, 73), (7, 13)] ...
280 0.792 [(0, 231771), (1, 185342), (2, 72960), (3, 19204), (4, 3674), (5, 546), (6, 89), (7, 6)] ...
290 0.789 [(0, 232200), (1, 184604), (2, 72817), (3, 18656), (4, 3681), (5, 618), (6, 75), (7, 10)] ...
300 0.788 [(0, 232527), (1, 183725), (2, 72792), (3, 18857), (4, 3666), (5, 560), (6, 60), (7, 8)] ...
310 0.787 [(0, 231805), (1, 183656), (2, 72170), (3, 18843), (4, 3552), (5, 570), (6, 76), (7, 13)] ...
320 0.785 [(0, 232322), (1, 183382), (2, 71595), (3, 18715), (4, 3654), (5, 524), (6, 74), (7, 3)] ...
330 0.782 [(0, 231791), (1, 183664), (2, 71459), (3, 18109), (4, 3547), (5, 498), (6, 57), (7, 4)] ...
340 0.781 [(0, 232610), (1, 182672), (2, 71087), (3, 18441), (4, 3523), (5, 516), (6, 63), (7, 9)] ...
350 0.782 [(0, 231450), (1, 182917), (2, 70927), (3, 18287), (4, 3535), (5, 584), (6, 53), (7, 9)] ...
360 0.778 [(0, 232209), (1, 182129), (2, 70526), (3, 18235), (4, 3435), (5, 524), (6, 54), (7, 4)] ...
370 0.779 [(0, 231881), (1, 182068), (2, 70979), (3, 17970), (4, 3492), (5, 504), (6, 66), (7, 8)] ...
380 0.774 [(0, 232532), (1, 181381), (2, 70038), (3, 17903), (4, 3403), (5, 520), (6, 55), (7, 0)] ...
390 0.776 [(0, 231769), (1, 181482), (2, 69997), (3, 17824), (4, 3433), (5, 535), (6, 72), (7, 4)] ...
400 0.775 [(0, 231970), (1, 180190), (2, 70510), (3, 17829), (4, 3311), (5, 510), (6, 60), (7, 7)] ...
410 0.771 [(0, 232139), (1, 180554), (2, 69684), (3, 17493), (4, 3308), (5, 519), (6, 62), (7, 7)] ...
420 0.771 [(0, 232061), (1, 180769), (2, 69196), (3, 17663), (4, 3318), (5, 496), (6, 60), (7, 8)] ...
430 0.768 [(0, 232488), (1, 179723), (2, 69088), (3, 17433), (4, 3238), (5, 459), (6, 72), (7, 4)] ...
440 0.770 [(0, 231710), (1, 180337), (2, 69286), (3, 17288), (4, 3295), (5, 499), (6, 57), (7, 10)] ...
450 0.765 [(0, 232510), (1, 179397), (2, 68669), (3, 17211), (4, 3162), (5, 499), (6, 52), (7, 5)] ...
460 0.768 [(0, 231624), (1, 179517), (2, 69107), (3, 17223), (4, 3180), (5, 452), (6, 57), (7, 9)] ...
470 0.766 [(0, 231624), (1, 179702), (2, 68400), (3, 17301), (4, 3149), (5, 455), (6, 63), (7, 10)] ...
480 0.765 [(0, 231984), (1, 178797), (2, 68078), (3, 17317), (4, 3224), (5, 477), (6, 60), (7, 6)] ...
490 0.763 [(0, 232274), (1, 178290), (2, 68176), (3, 16999), (4, 3201), (5, 500), (6, 53), (7, 6)] ...
500 0.760 [(0, 232256), (1, 178184), (2, 67589), (3, 16692), (4, 3144), (5, 460), (6, 54), (7, 4)] ...
510 0.764 [(0, 231271), (1, 178317), (2, 68147), (3, 17009), (4, 3179), (5, 459), (6, 46), (7, 7)] ...
520 0.761 [(0, 231858), (1, 177941), (2, 67705), (3, 16874), (4, 3100), (5, 462), (6, 65), (7, 5)] ...
530 0.762 [(0, 231514), (1, 177914), (2, 67579), (3, 16992), (4, 3199), (5, 421), (6, 58), (7, 6)] ...
540 0.759 [(0, 231606), (1, 177921), (2, 67129), (3, 16641), (4, 3141), (5, 471), (6, 62), (7, 4)] ...
550 0.757 [(0, 232107), (1, 177375), (2, 67105), (3, 16710), (4, 3023), (5, 417), (6, 52), (7, 5)] ...
560 0.757 [(0, 232199), (1, 176875), (2, 66795), (3, 16667), (4, 3102), (5, 435), (6, 60), (7, 2)] ...
570 0.758 [(0, 231941), (1, 176718), (2, 66777), (3, 16715), (4, 3247), (5, 422), (6, 60), (7, 7)] ...
580 0.755 [(0, 232488), (1, 176219), (2, 66487), (3, 16652), (4, 3025), (5, 446), (6, 49), (7, 3)] ...
590 0.755 [(0, 232044), (1, 176554), (2, 66460), (3, 16550), (4, 3073), (5, 424), (6, 49), (7, 4)] ...
600 0.753 [(0, 232457), (1, 176121), (2, 66236), (3, 16426), (4, 3008), (5, 423), (6, 66), (7, 3)] ...
610 0.752 [(0, 232118), (1, 175975), (2, 65956), (3, 16334), (4, 3078), (5, 414), (6, 64), (7, 3)] ...
620 0.752 [(0, 232339), (1, 175617), (2, 66010), (3, 16322), (4, 3056), (5, 434), (6, 56), (7, 7)] ...
630 0.754 [(0, 231364), (1, 175932), (2, 66112), (3, 16189), (4, 3098), (5, 469), (6, 49), (7, 5)] ...
640 0.752 [(0, 231490), (1, 175976), (2, 65881), (3, 16100), (4, 2981), (5, 445), (6, 65), (7, 7)] ...
650 0.751 [(0, 231960), (1, 175505), (2, 65570), (3, 16260), (4, 2952), (5, 479), (6, 47), (7, 2)] ...
660 0.750 [(0, 231829), (1, 175106), (2, 65704), (3, 15943), (4, 3043), (5, 448), (6, 69), (7, 7)] ...
670 0.750 [(0, 231496), (1, 175399), (2, 65428), (3, 16044), (4, 3009), (5, 414), (6, 53), (7, 0)] ...
680 0.747 [(0, 231967), (1, 175231), (2, 64846), (3, 16170), (4, 2893), (5, 427), (6, 42), (7, 4)] ...
690 0.748 [(0, 231724), (1, 174925), (2, 65247), (3, 15927), (4, 2908), (5, 392), (6, 58), (7, 3)] ...
700 0.748 [(0, 231834), (1, 174388), (2, 65250), (3, 16048), (4, 2947), (5, 448), (6, 55), (7, 4)] ...
710 0.746 [(0, 231771), (1, 174392), (2, 64726), (3, 15918), (4, 2839), (5, 437), (6, 52), (7, 6)] ...
720 0.746 [(0, 232141), (1, 173824), (2, 64886), (3, 15837), (4, 2935), (5, 441), (6, 52), (7, 5)] ...
730 0.747 [(0, 231415), (1, 173989), (2, 65040), (3, 15999), (4, 2886), (5, 439), (6, 44), (7, 4)] ...
740 0.742 [(0, 232247), (1, 173812), (2, 64234), (3, 15559), (4, 2885), (5, 396), (6, 57), (7, 4)] ...
750 0.744 [(0, 231584), (1, 173938), (2, 64673), (3, 15557), (4, 2797), (5, 432), (6, 55), (7, 2)] ...
760 0.744 [(0, 231445), (1, 173739), (2, 64806), (3, 15404), (4, 2908), (5, 396), (6, 57), (7, 5)] ...
770 0.743 [(0, 232064), (1, 173098), (2, 64313), (3, 15662), (4, 2841), (5, 458), (6, 54), (7, 7)] ...
780 0.744 [(0, 231575), (1, 173445), (2, 64679), (3, 15777), (4, 2825), (5, 411), (6, 58), (7, 7)] ...
790 0.743 [(0, 231061), (1, 173822), (2, 64129), (3, 15538), (4, 2885), (5, 423), (6, 57), (7, 10)] ...
800 0.742 [(0, 231650), (1, 173183), (2, 64165), (3, 15751), (4, 2806), (5, 394), (6, 55), (7, 3)] ...
810 0.741 [(0, 231682), (1, 173031), (2, 63810), (3, 15466), (4, 2846), (5, 434), (6, 43), (7, 3)] ...
820 0.740 [(0, 231586), (1, 172723), (2, 63901), (3, 15444), (4, 2784), (5, 412), (6, 49), (7, 3)] ...
830 0.739 [(0, 232079), (1, 172410), (2, 63859), (3, 15379), (4, 2806), (5, 414), (6, 53), (7, 5)] ...
840 0.740 [(0, 231694), (1, 172225), (2, 63717), (3, 15663), (4, 2803), (5, 405), (6, 45), (7, 4)] ...
850 0.738 [(0, 231392), (1, 172547), (2, 63505), (3, 15241), (4, 2721), (5, 412), (6, 51), (7, 5)] ...
860 0.739 [(0, 231460), (1, 172401), (2, 63403), (3, 15534), (4, 2783), (5, 413), (6, 53), (7, 1)] ...
870 0.736 [(0, 231991), (1, 171661), (2, 63410), (3, 15270), (4, 2679), (5, 412), (6, 47), (7, 3)] ...
880 0.737 [(0, 231258), (1, 172242), (2, 63006), (3, 15327), (4, 2785), (5, 384), (6, 42), (7, 5)] ...
890 0.736 [(0, 231426), (1, 171983), (2, 62928), (3, 15244), (4, 2787), (5, 390), (6, 39), (7, 6)] ...
900 0.734 [(0, 232550), (1, 171330), (2, 63014), (3, 15302), (4, 2694), (5, 411), (6, 36), (7, 5)] ...
910 0.733 [(0, 232182), (1, 172073), (2, 62469), (3, 15201), (4, 2665), (5, 380), (6, 50), (7, 6)] ...
920 0.736 [(0, 231302), (1, 171133), (2, 62988), (3, 15227), (4, 2765), (5, 394), (6, 37), (7, 4)] ...
930 0.736 [(0, 231389), (1, 171272), (2, 62875), (3, 15332), (4, 2681), (5, 402), (6, 34), (7, 5)] ...
940 0.735 [(0, 231605), (1, 171557), (2, 62354), (3, 15305), (4, 2768), (5, 400), (6, 56), (7, 0)] ...
950 0.733 [(0, 231559), (1, 171629), (2, 62555), (3, 14930), (4, 2770), (5, 362), (6, 37), (7, 3)] ...
960 0.732 [(0, 231979), (1, 170692), (2, 62653), (3, 15043), (4, 2703), (5, 362), (6, 48), (7, 2)] ...
970 0.731 [(0, 231902), (1, 170617), (2, 62065), (3, 15028), (4, 2768), (5, 364), (6, 38), (7, 0)] ...
980 0.730 [(0, 232077), (1, 170871), (2, 62165), (3, 14938), (4, 2630), (5, 399), (6, 43), (7, 6)] ...
990 0.731 [(0, 231942), (1, 170609), (2, 62264), (3, 14913), (4, 2666), (5, 368), (6, 55), (7, 8)] ...
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Old 2018-12-15, 23:01   #47
Batalov
 
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Quote:
Originally Posted by Dr Sardonicus View Post
I used the formula

round(exp(Euler)*log(p*log(2))/log(2))

on the first 50 (known) Mersenne prime exponents p, with the following result:
...
Quote:
Originally Posted by ewmayer View Post
Plotting your 'crowded run' the same way I did the actual M-prime exponents, we now see the largest dozen exponents again forming a 'hockey stick', ...
Quote:
Originally Posted by JeppeSN View Post
Yeah, if you try with 10M, that is:

(exp(Euler)/log(2))*log(log(2.0^10e6))

you get 40.47, which is too much...
Quote:
...and more...
The following explanations are too important to miss --
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Old 2018-12-15, 23:13   #48
Dr Sardonicus
 
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Quote:
Originally Posted by Batalov View Post
The following explanations are too important to miss --
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Old 2018-12-15, 23:26   #49
ewmayer
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So my highly rigorous piecewise-hockey-stick regression says "I always secretly wanted to be a Canadian?" Perhaps that explains my closet Anne of Green Gables fandom...

(Please don't tell my hockey buds, if they ever found out about that they'd throw me off the team, or worse, give me the dreaded "Zamboni haircut".)
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Old 2018-12-16, 00:26   #50
chalsall
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Quote:
Originally Posted by ewmayer View Post
So my highly rigorous piecewise-hockey-stick regression says "I always secretly wanted to be a Canadian?" Perhaps that explains my closet Anne of Green Gables fandom...
LOL... I like to joke that I was thrown out of Canada because I didn't like hockey....

(In truth, I didn't like the cold. Being 13.2 degrees above the equator, there's something quite nice about being able to wear shorts and sandals in December. Or any-time of the year, for that matter.)
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Old 2018-12-16, 18:55   #51
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Quote:
Originally Posted by Prime95 View Post
The 13th Mersenne prime between 10,000,000 and 100,000,000 -- expected 6.

For comparison:
1 to 10: 4
10 to 100: 6
100 to 1000: 4
1000 to 10000: 8
10000 to 100000: 6
100000 to 1000000: 5
1000000 to 10000000: 5
That's a funny way to bin the Mersenne prime exponents.

An alternative binning:

2 to 4: 2
4 to 8: 2
8 to 16: 1
16 to 32: 3
32 to 64: 1
64 to 128: 3
128 to 256: 0
256 to 512: 0
512 to 1024: 2
1024 to 2048: 1
2048 to 4096: 3
4096 to 8192: 2
8192 to 16384: 3
16384 to 32768: 3
32768 to 65536: 1
65536 to 131072: 2
131072 to 262144: 2
262144 to 524288: 0
524288 to 1.049M: 2
1.049M to 2.097M: 2
2.097M to 4.194M: 2
4.194M to 8.389M: 1
8.389M to 16.78M: 1
16.78M to 33.55M: 5
33.55M to 67.11M: 4 (...pending verification)
67.11M to 134.2M: 3 (...so far)
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Old 2018-12-16, 19:21   #52
GP2
 
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Quote:
Originally Posted by masser View Post
That's a funny way to bin the Mersenne prime exponents.
Asuming M51 < 268M, we have:

Code:
 5         0x1 –       0x10
 7        0x10 –      0x100
 6       0x100 –     0x1000
 9      0x1000 –    0x10000
 6     0x10000 –   0x100000
 6    0x100000 –  0x1000000
12   0x1000000 – 0x10000000   (so far)
All counts are final except for the last range above.

Assuming M51 < 89.5M, we are not even one-third of the way through the last range linearly, and not even 60% logarithmically.

Last fiddled with by GP2 on 2018-12-16 at 19:35
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Old 2018-12-16, 20:12   #53
GP2
 
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Not including M51, here's a histogram of the final hexadecimal digit of the exponents of Mersenne primes:

Code:
     #
     #       #
     #       #
     #       #
     #     # # # #
     #   # # # # #
     #   # # # # # #
     # # # # # # # #
     # # # # # # # #
 #   # # # # # # # #
 2   1 3 5 7 9 B D F
We can see that the prevalence of 1 (mod 4) vs. 3 (mod 4) is really a two-to-one prevalence of 1 (mod 8) vs. the other three:

(We include M51 here since its exponent has already been revealed to be 5 (mod 8) )

Code:
1    19
3     9
5    11 12
7    10

2     1
Similarly, a histogram of the final hexadecimal digit of the exponents of Wagstaff primes:

Code:
           #       #
           #       #
           #       #
       #   #       #
       #   # #     #
       #   # # # # #
     # # # # # # # #
     # # # # # # # #
     # # # # # # # #
     1 3 5 7 9 B D F
We can see that the prevalence of 3 (mod 4) vs. 1 (mod 4) is really a two-to-one prevalence of 7 (mod 8) vs. the other three:

Code:
1   8
3  10
5   7
7  18

Last fiddled with by GP2 on 2018-12-16 at 20:35 Reason: "prevalence" instead of "preference"
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Old 2018-12-16, 20:46   #54
preda
 
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Quote:
Originally Posted by GP2 View Post
Not including M51, here's a histogram of the final hexadecimal digit of the exponents of Mersenne primes:

Code:
     #
     #       #
     #       #
     #       #
     #     # # # #
     #   # # # # #
     #   # # # # # #
     # # # # # # # #
     # # # # # # # #
 #   # # # # # # # #
 2   1 3 5 7 9 B D F
We can see that the prevalence of 1 (mod 4) vs. 3 (mod 4) is really a two-to-one prevalence of 1 (mod 8) vs. the other three:

(We include M51 here since its exponent has already been revealed to be 5 (mod 8) )

Code:
1    19
3     9
5    11 12
7    10

2     1
Interesting. I wonder what is the histogram of not-yet-factored (TF or P-1) exponents. Are the unfactored exponents more uniformly distributed, and only MPs skewed, or is it the other way around -- the skew shape is inherited from the distribution of the unfactored population.

The secondary question is whether we should prioritize the testing of 1mod8 exponents.

Last fiddled with by preda on 2018-12-16 at 20:50
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Old 2018-12-16, 23:58   #55
Prime95
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For fun, I tweaked wagstaff.c to print the exponents where we "expect" to find the n-th Mersenne prime.

For exponents below 1 billion, Wagstaff et.al. predict just over 57 Mp:

Code:
Predicted Mp: 2, 3, 5, 7, 11, 17, 23, 31, 43, 61, 89, 127, 173, 241, 347, 467, 659, 937, 1297, 1831, 2591, 3617, 5099, 7193, 10133, 14389, 20297, 28687, 40801, 57709, 82007, 116747, 166063, 236479, 336857, 480787, 686639, 981077, 1402361, 2006651, 2873081, 4116617, 5902733, 8467903, 12152233, 17452667, 25078883, 36057451, 51858887, 74631061, 107448193, 154772603, 223029773, 321520513, 463692847, 669004691, 965590643
Final expected Mp: 57.0954
I think we're above the expected number for the first time in 500 years. The 127 to 521 and 216091 to 756839 gaps really set Mersenne searchers back.
Attached Files
File Type: c wagstaff.c (10.8 KB, 66 views)

Last fiddled with by Prime95 on 2018-12-16 at 23:59
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