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2018-12-14, 02:32
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#45 |
Feb 2017
Nowhere
13·359 Posts |
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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2018-12-14, 08:16
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#46 | |
Sep 2003
5×11×47 Posts |
Quote:
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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2018-12-15, 23:01
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#47 | ||||
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
100101000101102 Posts |
Quote:
Quote:
Quote:
Quote:
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2018-12-15, 23:13
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#48 | |
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Feb 2017
Nowhere
13×359 Posts |
Quote:
 
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2018-12-15, 23:26
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#49 |
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∂2ω=0
Sep 2002
República de California
19×613 Posts |
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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2018-12-16, 00:26
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#50 | |
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If I May
"Chris Halsall"
Sep 2002
Barbados
230478 Posts |
Quote:
(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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2018-12-16, 18:55
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#51 | |
Jul 2003
wear a mask
68C16 Posts |
Lucky 3
Quote:
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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2018-12-16, 19:21
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#52 | |
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Sep 2003
258510 Posts |
Quote:
Code:
5 0x1 – 0x10 7 0x10 – 0x100 6 0x100 – 0x1000 9 0x1000 – 0x10000 6 0x10000 – 0x100000 6 0x100000 – 0x1000000 12 0x1000000 – 0x10000000 (so far) 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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2018-12-16, 20:12
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#53 |
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Sep 2003
5×11×47 Posts |
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 include M51 here since its exponent has already been revealed to be 5 (mod 8) ) Code:
1 19 3 9 5 Code:
# #
# #
# #
# # #
# # # #
# # # # # #
# # # # # # # #
# # # # # # # #
# # # # # # # #
1 3 5 7 9 B D F
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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2018-12-16, 20:46
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#54 | |
"Mihai Preda"
Apr 2015
55B16 Posts |
Quote:
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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2018-12-16, 23:58
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#55 |
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P90 years forever!
Aug 2002
Yeehaw, FL
753710 Posts |
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 Last fiddled with by Prime95 on 2018-12-16 at 23:59 |
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| Thread | Thread Starter | Forum | Replies | Last Post |
| 500€ Reward for a proof for the Wagstaff primality test conjecture | Tony Reix | Wagstaff PRP Search | 7 | 2013-10-10 01:23 |
| Torture and Benchmarking test of Prome95.Doubts in implementation | paramveer | Information & Answers | 32 | 2012-01-15 06:05 |
| Wagstaff Conjecture | davieddy | Miscellaneous Math | 209 | 2011-01-23 23:50 |
| Algorithmic breakthroughs | davieddy | PrimeNet | 17 | 2010-07-21 00:07 |
| Conspiracy theories | jasong | Soap Box | 11 | 2010-07-05 12:23 |
