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2018-12-17, 01:31
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#56 | |
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
5,419 Posts |
Quote:
(Time frame extrapolation at https://www.mersenneforum.org/showpo...5&postcount=11) |
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2018-12-17, 02:58
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#57 | |
"Mihai Preda"
Apr 2015
3×457 Posts |
Quote:
As in: the probability of me winning the lottery tomorrow is not affected by the fact that I won the lottery today. Let's say, even if in average people win the lottery "once in 100years", the fact that I already won 10times does not change (does not diminish) my chance of winning again tomorrow. Or, better yet, if somehow I throw a fair coin 10times in a row heads, that does not imply that on the next throw I have a diminished chance of "heads" because of the expected average of a fair coin. Last fiddled with by preda on 2018-12-17 at 02:59 |
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2018-12-17, 04:41
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#58 | |
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"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
541910 Posts |
Quote:
I'm somewhat uncomfortable with applying probabilities to something as deterministic as is a number (now, previously, and forever, no coin toss or roulette wheel involved in setting the number's nature) prime or not, times n determinations on the set of remaining different prime exponents. The outcome of n accurate primality tests should be completely deterministic, and within the limits of hardware and software reliability, entirely repeatable, excluding effects of implementation-based things like pseudorandom fft shift on roundoff error. A straight sieving correctly performed would be entirely deterministic. You correctly describe the distribution as modeled. As a former employer would say, the map is not the territory. However useful it may be to have a map or model, it's not the same. We know that the primality of natural numbers are not independent. If n is prime, n>2, n+1 is composite, as a very simple example. On the bright side, when I make the case we should prepare ourselves for a long (multiyear) period between Mersenne prime discoveries, the search often goes better than that. Last fiddled with by kriesel on 2018-12-17 at 04:58 |
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2018-12-17, 06:24
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#59 |
Sep 2003
5·11·47 Posts |
As mentioned before, there are good reasons to believe that we have already discovered all factors of size 65 bits or less of all Mersenne numbers with prime exponent less than 1 billion. (We can expect TJAOI to complete the 66-bit level by August 2019).
Therefore this subset of all known factors is unchanging and complete, and can be suitable for various forms of analysis without any selection bias. In the table below: The 1st column is the starting point of the exponent range, in millions. Obviously we only consider prime exponents. The 2nd column is the average number of factors (of size 65 bits or less) of an exponent in this exponent range. The 3rd column is the total number of exponents in this exponent range. The subsequent columns show how many exponents in this range have 0, 1, 2, 3, 4, 5 or 6 factors (of size 65 bits or less). In each range there are a few exponents with 7 or more such factors, but this is omitted in the display table. We might expect the columns to form a Poisson distribution where k is the number of factors and the second column represents λ. Based on the heuristics in the 1983 paper by Wagstaff, we expect exponents with p=1 (mod 4) to have slightly fewer factors than exponents with p=3 (mod 4), and we see that below. (Divisors of Mersenne numbers ; Math. Comp. 40 (1983), 385-397 ; https://doi.org/10.1090/S0025-5718-1983-0679454-X ) However, neither those heuristics nor the data in the table below can explain the much stronger prevalence of p=1 (mod 4) vs. p=3 (mod 4) among the 50 known Mersenne primes with odd exponent: 31 of them are p=1, and 19 of them are p=3. Code:
******** For exponents with p = 2 (mod 4) 000 0.0000 1 [(0, 1)] ******** For exponents with p = 1 (mod 4) 000 1.0152 332180 [(0, 120212), (1, 122532), (2, 61818), (3, 21017), (4, 5273), (5, 1099), (6, 195)] ... 010 0.9229 302990 [(0, 119978), (1, 111570), (2, 51420), (3, 15736), (4, 3542), (5, 639), (6, 88)] ... 020 0.8993 293609 [(0, 119111), (1, 107759), (2, 48503), (3, 14406), (4, 3188), (5, 558), (6, 74)] ... 030 0.8794 287908 [(0, 119039), (1, 105783), (2, 45990), (3, 13634), (4, 2890), (5, 488), (6, 69)] ... 040 0.8608 283765 [(0, 119754), (1, 103583), (2, 44414), (3, 12765), (4, 2759), (5, 418), (6, 66)] ... 050 0.8561 280218 [(0, 118645), (1, 102342), (2, 43836), (3, 12264), (4, 2626), (5, 444), (6, 52)] ... 060 0.8462 278048 [(0, 118877), (1, 101523), (2, 42676), (3, 11995), (4, 2536), (5, 381), (6, 54)] ... 070 0.8379 275655 [(0, 118771), (1, 100479), (2, 42016), (3, 11598), (4, 2349), (5, 392), (6, 47)] ... 080 0.8287 274020 [(0, 119116), (1, 99895), (2, 41031), (3, 11303), (4, 2226), (5, 391), (6, 50)] ... 090 0.8247 272111 [(0, 119052), (1, 98674), (2, 40528), (3, 11228), (4, 2209), (5, 370), (6, 45)] ... 100 0.8200 270854 [(0, 119277), (1, 97805), (2, 40087), (3, 11072), (4, 2229), (5, 340), (6, 32)] ... 110 0.8146 269117 [(0, 118643), (1, 97812), (2, 39589), (3, 10548), (4, 2112), (5, 352), (6, 56)] ... 120 0.8132 268389 [(0, 118642), (1, 97280), (2, 39395), (3, 10535), (4, 2173), (5, 315), (6, 42)] ... 130 0.8100 267056 [(0, 118414), (1, 96779), (2, 39002), (3, 10361), (4, 2109), (5, 332), (6, 49)] ... 140 0.8032 266064 [(0, 118936), (1, 95887), (2, 38674), (3, 10197), (4, 2020), (5, 306), (6, 38)] ... 150 0.8023 264739 [(0, 118160), (1, 95821), (2, 38400), (3, 10058), (4, 1946), (5, 312), (6, 36)] ... 160 0.7970 264226 [(0, 118717), (1, 95378), (2, 37800), (3, 10081), (4, 1920), (5, 288), (6, 36)] ... 170 0.7983 263745 [(0, 118226), (1, 95421), (2, 37804), (3, 10056), (4, 1894), (5, 296), (6, 43)] ... 180 0.7949 262664 [(0, 118482), (1, 94497), (2, 37494), (3, 9858), (4, 1986), (5, 301), (6, 40)] ... 190 0.7901 261462 [(0, 118347), (1, 94226), (2, 36974), (3, 9664), (4, 1898), (5, 301), (6, 48)] ... 200 0.7842 261268 [(0, 118945), (1, 93783), (2, 36953), (3, 9480), (4, 1802), (5, 274), (6, 26)] ... 210 0.7868 260763 [(0, 118426), (1, 93748), (2, 36811), (3, 9661), (4, 1806), (5, 271), (6, 37)] ... 220 0.7828 259931 [(0, 118565), (1, 93368), (2, 36473), (3, 9311), (4, 1878), (5, 297), (6, 33)] ... 230 0.7781 259477 [(0, 118751), (1, 93317), (2, 36091), (3, 9240), (4, 1760), (5, 275), (6, 37)] ... 240 0.7788 258715 [(0, 118349), (1, 92997), (2, 36029), (3, 9308), (4, 1705), (5, 284), (6, 42)] ... 250 0.7780 258403 [(0, 118461), (1, 92649), (2, 35963), (3, 9250), (4, 1757), (5, 270), (6, 48)] ... 260 0.7761 258126 [(0, 118567), (1, 92505), (2, 35778), (3, 9220), (4, 1729), (5, 286), (6, 37)] ... 270 0.7752 257244 [(0, 118461), (1, 91846), (2, 35685), (3, 9179), (4, 1763), (5, 270), (6, 33)] ... 280 0.7759 256730 [(0, 117902), (1, 92028), (2, 35602), (3, 9167), (4, 1742), (5, 247), (6, 41)] ... 290 0.7700 256241 [(0, 118610), (1, 91515), (2, 35054), (3, 8954), (4, 1774), (5, 294), (6, 34)] ... 300 0.7707 256288 [(0, 118497), (1, 91365), (2, 35465), (3, 8940), (4, 1734), (5, 265), (6, 17)] ... 310 0.7690 255394 [(0, 118102), (1, 91408), (2, 34975), (3, 8953), (4, 1660), (5, 251), (6, 40)] ... 320 0.7663 254928 [(0, 118407), (1, 90886), (2, 34735), (3, 8896), (4, 1738), (5, 234), (6, 32)] ... 330 0.7632 254683 [(0, 118347), (1, 91163), (2, 34536), (3, 8692), (4, 1690), (5, 225), (6, 29)] ... 340 0.7613 254161 [(0, 118365), (1, 90866), (2, 34317), (3, 8740), (4, 1634), (5, 201), (6, 34)] ... 350 0.7656 253755 [(0, 117855), (1, 90650), (2, 34424), (3, 8847), (4, 1691), (5, 261), (6, 24)] ... 360 0.7599 253710 [(0, 118354), (1, 90528), (2, 34347), (3, 8656), (4, 1556), (5, 244), (6, 24)] ... 370 0.7590 253403 [(0, 118345), (1, 90378), (2, 34241), (3, 8576), (4, 1599), (5, 232), (6, 28)] ... 380 0.7559 253077 [(0, 118639), (1, 90175), (2, 33835), (3, 8559), (4, 1601), (5, 241), (6, 26)] ... 390 0.7571 252490 [(0, 118123), (1, 90175), (2, 33842), (3, 8434), (4, 1624), (5, 264), (6, 25)] ... 400 0.7542 252255 [(0, 118514), (1, 89424), (2, 34196), (3, 8339), (4, 1527), (5, 227), (6, 24)] ... 410 0.7542 251924 [(0, 118274), (1, 89627), (2, 33848), (3, 8336), (4, 1572), (5, 233), (6, 32)] ... 420 0.7544 251590 [(0, 118026), (1, 89648), (2, 33732), (3, 8341), (4, 1578), (5, 239), (6, 25)] ... 430 0.7498 251025 [(0, 118361), (1, 89018), (2, 33731), (3, 8187), (4, 1494), (5, 194), (6, 37)] ... 440 0.7513 251535 [(0, 118405), (1, 89461), (2, 33629), (3, 8207), (4, 1568), (5, 227), (6, 32)] ... 450 0.7476 250626 [(0, 118200), (1, 89381), (2, 33186), (3, 8106), (4, 1496), (5, 230), (6, 22)] ... 460 0.7505 250429 [(0, 117838), (1, 89132), (2, 33605), (3, 8093), (4, 1510), (5, 224), (6, 21)] ... 470 0.7469 250328 [(0, 118193), (1, 89184), (2, 33029), (3, 8247), (4, 1427), (5, 207), (6, 35)] ... 480 0.7465 249898 [(0, 118291), (1, 88608), (2, 33068), (3, 8194), (4, 1489), (5, 221), (6, 25)] ... 490 0.7463 249591 [(0, 118181), (1, 88536), (2, 32964), (3, 8121), (4, 1538), (5, 227), (6, 20)] ... 500 0.7416 249413 [(0, 118510), (1, 88427), (2, 32795), (3, 8023), (4, 1425), (5, 210), (6, 21)] ... 510 0.7480 249280 [(0, 117800), (1, 88391), (2, 33159), (3, 8213), (4, 1494), (5, 202), (6, 16)] ... 520 0.7418 249065 [(0, 118347), (1, 88300), (2, 32711), (3, 8071), (4, 1395), (5, 212), (6, 26)] ... 530 0.7436 248897 [(0, 118158), (1, 88102), (2, 32885), (3, 8051), (4, 1491), (5, 182), (6, 25)] ... 540 0.7387 248580 [(0, 118364), (1, 88319), (2, 32333), (3, 7900), (4, 1417), (5, 215), (6, 28)] ... 550 0.7386 248341 [(0, 118335), (1, 87991), (2, 32474), (3, 7903), (4, 1427), (5, 187), (6, 22)] ... 560 0.7384 247962 [(0, 118180), (1, 88031), (2, 32136), (3, 7924), (4, 1463), (5, 201), (6, 26)] ... 570 0.7392 248316 [(0, 118431), (1, 87840), (2, 32403), (3, 7906), (4, 1522), (5, 182), (6, 30)] ... 580 0.7350 247663 [(0, 118698), (1, 87323), (2, 32110), (3, 7898), (4, 1408), (5, 203), (6, 22)] ... 590 0.7380 247605 [(0, 118160), (1, 87670), (2, 32163), (3, 7953), (4, 1449), (5, 187), (6, 21)] ... 600 0.7346 247249 [(0, 118476), (1, 87317), (2, 31960), (3, 7843), (4, 1430), (5, 194), (6, 27)] ... 610 0.7373 246725 [(0, 117915), (1, 87117), (2, 32254), (3, 7724), (4, 1503), (5, 179), (6, 32)] ... 620 0.7348 246873 [(0, 118224), (1, 87135), (2, 32093), (3, 7835), (4, 1376), (5, 189), (6, 17)] ... 630 0.7369 246609 [(0, 117684), (1, 87348), (2, 32269), (3, 7643), (4, 1444), (5, 203), (6, 16)] ... 640 0.7337 246608 [(0, 118104), (1, 87199), (2, 32058), (3, 7627), (4, 1400), (5, 186), (6, 33)] ... 650 0.7316 246038 [(0, 118042), (1, 87172), (2, 31553), (3, 7624), (4, 1401), (5, 232), (6, 14)] ... 660 0.7324 246119 [(0, 118289), (1, 86676), (2, 31867), (3, 7583), (4, 1461), (5, 206), (6, 37)] ... 670 0.7355 246091 [(0, 117651), (1, 87136), (2, 31963), (3, 7678), (4, 1443), (5, 199), (6, 20)] ... 680 0.7290 245793 [(0, 118326), (1, 86794), (2, 31405), (3, 7708), (4, 1356), (5, 188), (6, 15)] ... 690 0.7291 245481 [(0, 118155), (1, 86683), (2, 31476), (3, 7573), (4, 1385), (5, 181), (6, 26)] ... 700 0.7299 245622 [(0, 118226), (1, 86584), (2, 31609), (3, 7585), (4, 1389), (5, 204), (6, 22)] ... 710 0.7266 245171 [(0, 118232), (1, 86502), (2, 31453), (3, 7447), (4, 1316), (5, 194), (6, 24)] ... 720 0.7266 244938 [(0, 118206), (1, 86279), (2, 31459), (3, 7446), (4, 1317), (5, 205), (6, 26)] ... 730 0.7296 244738 [(0, 117833), (1, 86206), (2, 31517), (3, 7643), (4, 1323), (5, 195), (6, 19)] ... 740 0.7270 244396 [(0, 117892), (1, 86232), (2, 31194), (3, 7504), (4, 1359), (5, 191), (6, 22)] ... 750 0.7265 244705 [(0, 118039), (1, 86388), (2, 31265), (3, 7454), (4, 1328), (5, 206), (6, 23)] ... 760 0.7256 244807 [(0, 118018), (1, 86500), (2, 31518), (3, 7227), (4, 1329), (5, 191), (6, 21)] ... 770 0.7237 244132 [(0, 118332), (1, 85671), (2, 31214), (3, 7343), (4, 1344), (5, 200), (6, 26)] ... 780 0.7274 244460 [(0, 117872), (1, 86037), (2, 31604), (3, 7469), (4, 1272), (5, 171), (6, 32)] ... 790 0.7255 243975 [(0, 117583), (1, 86375), (2, 31171), (3, 7337), (4, 1294), (5, 189), (6, 19)] ... 800 0.7254 244097 [(0, 118032), (1, 85933), (2, 31048), (3, 7565), (4, 1293), (5, 195), (6, 29)] ... 810 0.7244 243536 [(0, 117913), (1, 85563), (2, 31131), (3, 7377), (4, 1317), (5, 213), (6, 20)] ... 820 0.7202 243656 [(0, 118302), (1, 85700), (2, 30833), (3, 7383), (4, 1250), (5, 170), (6, 18)] ... 830 0.7220 243486 [(0, 118020), (1, 85767), (2, 30805), (3, 7372), (4, 1322), (5, 175), (6, 24)] ... 840 0.7233 243081 [(0, 117853), (1, 85418), (2, 30815), (3, 7446), (4, 1329), (5, 203), (6, 15)] ... 850 0.7207 243037 [(0, 117767), (1, 85957), (2, 30526), (3, 7269), (4, 1280), (5, 209), (6, 26)] ... 860 0.7243 242782 [(0, 117445), (1, 85590), (2, 30783), (3, 7422), (4, 1310), (5, 201), (6, 31)] ... 870 0.7192 242938 [(0, 118238), (1, 85159), (2, 30795), (3, 7272), (4, 1255), (5, 192), (6, 25)] ... 880 0.7187 242350 [(0, 117888), (1, 85246), (2, 30439), (3, 7289), (4, 1285), (5, 175), (6, 24)] ... 890 0.7204 242536 [(0, 117603), (1, 85718), (2, 30420), (3, 7243), (4, 1335), (5, 196), (6, 19)] ... 900 0.7145 242910 [(0, 118873), (1, 84791), (2, 30619), (3, 7195), (4, 1241), (5, 171), (6, 18)] ... 910 0.7155 242624 [(0, 118434), (1, 85167), (2, 30354), (3, 7200), (4, 1266), (5, 172), (6, 27)] ... 920 0.7205 241813 [(0, 117625), (1, 84725), (2, 30654), (3, 7262), (4, 1341), (5, 190), (6, 14)] ... 930 0.7210 242235 [(0, 117740), (1, 84975), (2, 30609), (3, 7418), (4, 1281), (5, 193), (6, 15)] ... 940 0.7191 242177 [(0, 117781), (1, 85187), (2, 30427), (3, 7270), (4, 1305), (5, 175), (6, 31)] ... 950 0.7175 241702 [(0, 117544), (1, 85039), (2, 30641), (3, 7030), (4, 1258), (5, 171), (6, 16)] ... 960 0.7142 241850 [(0, 118124), (1, 84915), (2, 30221), (3, 7167), (4, 1253), (5, 151), (6, 19)] ... 970 0.7153 241371 [(0, 118063), (1, 84337), (2, 30313), (3, 7171), (4, 1292), (5, 172), (6, 23)] ... 980 0.7129 241561 [(0, 118240), (1, 84655), (2, 30084), (3, 7159), (4, 1226), (5, 178), (6, 16)] ... 990 0.7137 241289 [(0, 118130), (1, 84436), (2, 30076), (3, 7179), (4, 1271), (5, 177), (6, 19)] ... ******** For exponents with p = 3 (mod 4) 000 1.0599 332398 [(0, 114506), (1, 122927), (2, 64753), (3, 22756), (4, 5993), (5, 1186), (6, 239)] ... 010 0.9691 303038 [(0, 114718), (1, 111577), (2, 54225), (3, 17468), (4, 4152), (5, 765), (6, 116)] ... 020 0.9358 293643 [(0, 114611), (1, 108364), (2, 50821), (3, 15500), (4, 3590), (5, 645), (6, 91)] ... 030 0.9187 287887 [(0, 114200), (1, 106406), (2, 48621), (3, 14664), (4, 3269), (5, 603), (6, 108)] ... 040 0.9085 283715 [(0, 113773), (1, 104537), (2, 47578), (3, 14032), (4, 3114), (5, 592), (6, 81)] ... 050 0.8942 280763 [(0, 114118), (1, 103300), (2, 46445), (3, 13383), (4, 2933), (5, 516), (6, 60)] ... 060 0.8857 277901 [(0, 114063), (1, 102191), (2, 45080), (3, 13147), (4, 2838), (5, 511), (6, 62)] ... 070 0.8781 275663 [(0, 113923), (1, 101225), (2, 44494), (3, 12825), (4, 2683), (5, 452), (6, 54)] ... 080 0.8710 273552 [(0, 113932), (1, 100403), (2, 43569), (3, 12417), (4, 2752), (5, 421), (6, 51)] ... 090 0.8649 272390 [(0, 114342), (1, 99657), (2, 42895), (3, 12422), (4, 2572), (5, 436), (6, 58)] ... 100 0.8585 271000 [(0, 114214), (1, 99527), (2, 42251), (3, 11982), (4, 2519), (5, 447), (6, 53)] ... 110 0.8563 269222 [(0, 113782), (1, 98725), (2, 41828), (3, 11954), (4, 2451), (5, 411), (6, 61)] ... 120 0.8509 268150 [(0, 114190), (1, 97748), (2, 41697), (3, 11588), (4, 2448), (5, 409), (6, 62)] ... 130 0.8484 266956 [(0, 113618), (1, 97700), (2, 41454), (3, 11391), (4, 2350), (5, 374), (6, 61)] ... 140 0.8433 266133 [(0, 113951), (1, 97188), (2, 40961), (3, 11310), (4, 2277), (5, 396), (6, 46)] ... 150 0.8378 265323 [(0, 114048), (1, 97288), (2, 40167), (3, 11098), (4, 2304), (5, 362), (6, 49)] ... 160 0.8363 264399 [(0, 113697), (1, 96983), (2, 40125), (3, 10982), (4, 2176), (5, 385), (6, 46)] ... 170 0.8326 263557 [(0, 114295), (1, 95776), (2, 39910), (3, 10924), (4, 2266), (5, 326), (6, 52)] ... 180 0.8320 262424 [(0, 113602), (1, 95734), (2, 39743), (3, 10753), (4, 2168), (5, 378), (6, 40)] ... 190 0.8309 262002 [(0, 113579), (1, 95417), (2, 39823), (3, 10552), (4, 2208), (5, 385), (6, 34)] ... 200 0.8251 261421 [(0, 113902), (1, 95306), (2, 39138), (3, 10565), (4, 2170), (5, 294), (6, 41)] ... 210 0.8234 260151 [(0, 113789), (1, 94362), (2, 39008), (3, 10547), (4, 2065), (5, 344), (6, 33)] ... 220 0.8188 260102 [(0, 114289), (1, 94281), (2, 38687), (3, 10474), (4, 2012), (5, 321), (6, 35)] ... 230 0.8194 259494 [(0, 113915), (1, 94106), (2, 38782), (3, 10204), (4, 2126), (5, 322), (6, 34)] ... 240 0.8152 259059 [(0, 114084), (1, 94119), (2, 38358), (3, 10057), (4, 2088), (5, 307), (6, 39)] ... 250 0.8155 258139 [(0, 113480), (1, 94108), (2, 38058), (3, 10114), (4, 2002), (5, 328), (6, 42)] ... 260 0.8143 257398 [(0, 113585), (1, 93464), (2, 37699), (3, 10248), (4, 2059), (5, 298), (6, 43)] ... 270 0.8080 257441 [(0, 114004), (1, 93771), (2, 37506), (3, 9823), (4, 1985), (5, 305), (6, 40)] ... 280 0.8086 256864 [(0, 113869), (1, 93314), (2, 37358), (3, 10037), (4, 1932), (5, 299), (6, 48)] ... 290 0.8082 256421 [(0, 113590), (1, 93089), (2, 37763), (3, 9702), (4, 1907), (5, 324), (6, 41)] ... 300 0.8060 255908 [(0, 114030), (1, 92360), (2, 37327), (3, 9917), (4, 1932), (5, 295), (6, 43)] ... 310 0.8059 255291 [(0, 113703), (1, 92248), (2, 37195), (3, 9890), (4, 1892), (5, 319), (6, 36)] ... 320 0.8031 255341 [(0, 113915), (1, 92496), (2, 36860), (3, 9819), (4, 1916), (5, 290), (6, 42)] ... 330 0.8001 254447 [(0, 113444), (1, 92501), (2, 36923), (3, 9417), (4, 1857), (5, 273), (6, 28)] ... 340 0.7999 254760 [(0, 114245), (1, 91806), (2, 36770), (3, 9701), (4, 1889), (5, 315), (6, 29)] ... 350 0.7984 254007 [(0, 113595), (1, 92267), (2, 36503), (3, 9440), (4, 1844), (5, 323), (6, 29)] ... 360 0.7964 253407 [(0, 113855), (1, 91601), (2, 36179), (3, 9579), (4, 1879), (5, 280), (6, 30)] ... 370 0.7988 253565 [(0, 113536), (1, 91690), (2, 36738), (3, 9394), (4, 1893), (5, 272), (6, 38)] ... 380 0.7930 252760 [(0, 113893), (1, 91206), (2, 36203), (3, 9344), (4, 1802), (5, 279), (6, 29)] ... 390 0.7943 252626 [(0, 113646), (1, 91307), (2, 36155), (3, 9390), (4, 1809), (5, 271), (6, 47)] ... 400 0.7958 252132 [(0, 113456), (1, 90766), (2, 36314), (3, 9490), (4, 1784), (5, 283), (6, 36)] ... 410 0.7888 251842 [(0, 113865), (1, 90927), (2, 35836), (3, 9157), (4, 1736), (5, 286), (6, 30)] ... 420 0.7878 251981 [(0, 114035), (1, 91121), (2, 35464), (3, 9322), (4, 1740), (5, 257), (6, 35)] ... 430 0.7860 251480 [(0, 114127), (1, 90705), (2, 35357), (3, 9246), (4, 1744), (5, 265), (6, 35)] ... 440 0.7886 250948 [(0, 113305), (1, 90876), (2, 35657), (3, 9081), (4, 1727), (5, 272), (6, 25)] ... 450 0.7833 250884 [(0, 114310), (1, 90016), (2, 35483), (3, 9105), (4, 1666), (5, 269), (6, 30)] ... 460 0.7850 250741 [(0, 113786), (1, 90385), (2, 35502), (3, 9130), (4, 1670), (5, 228), (6, 36)] ... 470 0.7859 250378 [(0, 113431), (1, 90518), (2, 35371), (3, 9054), (4, 1722), (5, 248), (6, 28)] ... 480 0.7840 250046 [(0, 113693), (1, 90189), (2, 35010), (3, 9123), (4, 1735), (5, 256), (6, 35)] ... 490 0.7805 249908 [(0, 114093), (1, 89754), (2, 35212), (3, 8878), (4, 1663), (5, 273), (6, 33)] ... 500 0.7780 248972 [(0, 113746), (1, 89757), (2, 34794), (3, 8669), (4, 1719), (5, 250), (6, 33)] ... 510 0.7807 249155 [(0, 113471), (1, 89926), (2, 34988), (3, 8796), (4, 1685), (5, 257), (6, 30)] ... 520 0.7807 248946 [(0, 113511), (1, 89641), (2, 34994), (3, 8803), (4, 1705), (5, 250), (6, 39)] ... 530 0.7809 248786 [(0, 113356), (1, 89812), (2, 34694), (3, 8941), (4, 1708), (5, 239), (6, 33)] ... 540 0.7802 248395 [(0, 113242), (1, 89602), (2, 34796), (3, 8741), (4, 1724), (5, 256), (6, 34)] ... 550 0.7760 248453 [(0, 113772), (1, 89384), (2, 34631), (3, 8807), (4, 1596), (5, 230), (6, 30)] ... 560 0.7750 248174 [(0, 114019), (1, 88844), (2, 34659), (3, 8743), (4, 1639), (5, 234), (6, 34)] ... 570 0.7771 247573 [(0, 113510), (1, 88878), (2, 34374), (3, 8809), (4, 1725), (5, 240), (6, 30)] ... 580 0.7742 247706 [(0, 113790), (1, 88896), (2, 34377), (3, 8754), (4, 1617), (5, 243), (6, 27)] ... 590 0.7721 247554 [(0, 113884), (1, 88884), (2, 34297), (3, 8597), (4, 1624), (5, 237), (6, 28)] ... 600 0.7710 247492 [(0, 113981), (1, 88804), (2, 34276), (3, 8583), (4, 1578), (5, 229), (6, 39)] ... 610 0.7677 247219 [(0, 114203), (1, 88858), (2, 33702), (3, 8610), (4, 1575), (5, 235), (6, 32)] ... 620 0.7692 246968 [(0, 114115), (1, 88482), (2, 33917), (3, 8487), (4, 1680), (5, 245), (6, 39)] ... 630 0.7708 246612 [(0, 113680), (1, 88584), (2, 33843), (3, 8546), (4, 1654), (5, 266), (6, 33)] ... 640 0.7701 246337 [(0, 113386), (1, 88777), (2, 33823), (3, 8473), (4, 1581), (5, 259), (6, 32)] ... 650 0.7698 246737 [(0, 113918), (1, 88333), (2, 34017), (3, 8636), (4, 1551), (5, 247), (6, 33)] ... 660 0.7680 246030 [(0, 113540), (1, 88430), (2, 33837), (3, 8360), (4, 1582), (5, 242), (6, 32)] ... 670 0.7644 245757 [(0, 113845), (1, 88263), (2, 33465), (3, 8366), (4, 1566), (5, 215), (6, 33)] ... 680 0.7658 245787 [(0, 113641), (1, 88437), (2, 33441), (3, 8462), (4, 1537), (5, 239), (6, 27)] ... 690 0.7659 245703 [(0, 113569), (1, 88242), (2, 33771), (3, 8354), (4, 1523), (5, 211), (6, 32)] ... 700 0.7668 245352 [(0, 113608), (1, 87804), (2, 33641), (3, 8463), (4, 1558), (5, 244), (6, 33)] ... 710 0.7648 244972 [(0, 113539), (1, 87890), (2, 33273), (3, 8471), (4, 1523), (5, 243), (6, 28)] ... 720 0.7644 245184 [(0, 113935), (1, 87545), (2, 33427), (3, 8391), (4, 1618), (5, 236), (6, 26)] ... 730 0.7652 245079 [(0, 113582), (1, 87783), (2, 33523), (3, 8356), (4, 1563), (5, 244), (6, 25)] ... 740 0.7565 244798 [(0, 114355), (1, 87580), (2, 33040), (3, 8055), (4, 1526), (5, 205), (6, 35)] ... 750 0.7607 244333 [(0, 113545), (1, 87550), (2, 33408), (3, 8103), (4, 1469), (5, 226), (6, 32)] ... 760 0.7621 243953 [(0, 113427), (1, 87239), (2, 33288), (3, 8177), (4, 1579), (5, 205), (6, 36)] ... 770 0.7614 244365 [(0, 113732), (1, 87427), (2, 33099), (3, 8319), (4, 1497), (5, 258), (6, 28)] ... 780 0.7616 244317 [(0, 113703), (1, 87408), (2, 33075), (3, 8308), (4, 1553), (5, 240), (6, 26)] ... 790 0.7614 243951 [(0, 113478), (1, 87447), (2, 32958), (3, 8201), (4, 1591), (5, 234), (6, 38)] ... 800 0.7595 243911 [(0, 113618), (1, 87250), (2, 33117), (3, 8186), (4, 1513), (5, 199), (6, 26)] ... 810 0.7567 243779 [(0, 113769), (1, 87468), (2, 32679), (3, 8089), (4, 1529), (5, 221), (6, 23)] ... 820 0.7601 243246 [(0, 113284), (1, 87023), (2, 33068), (3, 8061), (4, 1534), (5, 242), (6, 31)] ... 830 0.7560 243520 [(0, 114059), (1, 86643), (2, 33054), (3, 8007), (4, 1484), (5, 239), (6, 29)] ... 840 0.7572 243475 [(0, 113841), (1, 86807), (2, 32902), (3, 8217), (4, 1474), (5, 202), (6, 30)] ... 850 0.7553 242837 [(0, 113625), (1, 86590), (2, 32979), (3, 7972), (4, 1441), (5, 203), (6, 25)] ... 860 0.7542 243266 [(0, 114015), (1, 86811), (2, 32620), (3, 8112), (4, 1473), (5, 212), (6, 22)] ... 870 0.7531 242535 [(0, 113753), (1, 86502), (2, 32615), (3, 7998), (4, 1424), (5, 220), (6, 22)] ... 880 0.7557 242701 [(0, 113370), (1, 86996), (2, 32567), (3, 8038), (4, 1500), (5, 209), (6, 18)] ... 890 0.7521 242267 [(0, 113823), (1, 86265), (2, 32508), (3, 8001), (4, 1452), (5, 194), (6, 20)] ... 900 0.7540 242432 [(0, 113677), (1, 86539), (2, 32395), (3, 8107), (4, 1453), (5, 240), (6, 18)] ... 910 0.7505 242402 [(0, 113748), (1, 86906), (2, 32115), (3, 8001), (4, 1399), (5, 208), (6, 23)] ... 920 0.7513 242037 [(0, 113677), (1, 86408), (2, 32334), (3, 7965), (4, 1424), (5, 204), (6, 23)] ... 930 0.7501 241756 [(0, 113649), (1, 86297), (2, 32266), (3, 7914), (4, 1400), (5, 209), (6, 19)] ... 940 0.7503 241870 [(0, 113824), (1, 86370), (2, 31927), (3, 8035), (4, 1463), (5, 225), (6, 25)] ... 950 0.7485 242143 [(0, 114015), (1, 86590), (2, 31914), (3, 7900), (4, 1512), (5, 191), (6, 21)] ... 960 0.7504 241632 [(0, 113855), (1, 85777), (2, 32432), (3, 7876), (4, 1450), (5, 211), (6, 29)] ... 970 0.7469 241411 [(0, 113839), (1, 86280), (2, 31752), (3, 7857), (4, 1476), (5, 192), (6, 15)] ... 980 0.7477 241568 [(0, 113837), (1, 86216), (2, 32081), (3, 7779), (4, 1404), (5, 221), (6, 27)] ... 990 0.7475 241536 [(0, 113812), (1, 86173), (2, 32188), (3, 7734), (4, 1395), (5, 191), (6, 36)] ... |
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2018-12-17, 06:40
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#60 | |
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Sep 2003
5×11×47 Posts |
Quote:
With Wagstaff primes there is an analogous situation. There is a prevalence of p=3 (mod 4) vs. p=1 (mod 4) among the 43 known Wagstaff primes: 15 of them are p=1 and 28 of them are p=3. However, this is really a two-to-one prevalence of p=7 (mod 8) vs. each of p=1, p=3, p=5 (mod 8): the numbers are 8, 10, 7, 18, respectively. We can look at the table of factor frequencies again (for factors of size 65 bits or less), this time separating out the data for each of the congruence classes modulo 8. However, once again this doesn't really tell us anything about the observed prevalences for Mersenne primes. In the table below, the frequencies for p=1 (mod 4) are split fairly evenly between p=1 (mod 8) and p=5 (mod 8). Likewise, the frequencies for p=3 (mod 4) are split fairly evenly between p=3 (mod 8) and p=7 (mod 8). So there isn't much additional information here. The table for mod 4 and an explanation of the various columns was in the previous post. Code:
******** For exponents with p = 2 (mod 8) 000 0.0000 1 [(0, 1)] ... ******** For exponents with p = 1 (mod 8) 000 1.0172 165976 [(0, 59835), (1, 61374), (2, 30944), (3, 10513), (4, 2645), (5, 549), (6, 101)] ... 010 0.9240 151501 [(0, 60092), (1, 55571), (2, 25715), (3, 7965), (4, 1772), (5, 325), (6, 53)] ... 020 0.9009 146811 [(0, 59459), (1, 53897), (2, 24259), (3, 7306), (4, 1579), (5, 266), (6, 40)] ... 030 0.8825 143812 [(0, 59241), (1, 52866), (2, 23139), (3, 6837), (4, 1444), (5, 237), (6, 37)] ... 040 0.8607 141919 [(0, 59818), (1, 51932), (2, 22169), (3, 6397), (4, 1366), (5, 202), (6, 31)] ... 050 0.8571 139947 [(0, 59314), (1, 51001), (2, 21843), (3, 6198), (4, 1319), (5, 242), (6, 26)] ... 060 0.8448 139016 [(0, 59519), (1, 50695), (2, 21353), (3, 6004), (4, 1232), (5, 183), (6, 25)] ... 070 0.8411 137817 [(0, 59143), (1, 50305), (2, 21143), (3, 5822), (4, 1190), (5, 186), (6, 26)] ... 080 0.8304 136986 [(0, 59289), (1, 50220), (2, 20511), (3, 5618), (4, 1123), (5, 194), (6, 26)] ... 090 0.8263 136185 [(0, 59534), (1, 49292), (2, 20375), (3, 5672), (4, 1105), (5, 188), (6, 18)] ... 100 0.8217 135291 [(0, 59532), (1, 48846), (2, 19987), (3, 5582), (4, 1147), (5, 173), (6, 15)] ... 110 0.8141 134662 [(0, 59372), (1, 48968), (2, 19836), (3, 5183), (4, 1103), (5, 175), (6, 24)] ... 120 0.8113 134352 [(0, 59569), (1, 48599), (2, 19673), (3, 5213), (4, 1102), (5, 170), (6, 23)] ... 130 0.8100 133412 [(0, 59171), (1, 48332), (2, 19480), (3, 5165), (4, 1068), (5, 169), (6, 24)] ... 140 0.8036 132900 [(0, 59470), (1, 47814), (2, 19291), (3, 5113), (4, 1025), (5, 165), (6, 20)] ... 150 0.8039 132582 [(0, 59071), (1, 47908), (2, 19435), (3, 5054), (4, 948), (5, 144), (6, 20)] ... 160 0.7971 132082 [(0, 59377), (1, 47675), (2, 18839), (3, 5036), (4, 982), (5, 149), (6, 20)] ... 170 0.7987 131896 [(0, 59072), (1, 47790), (2, 18854), (3, 5070), (4, 933), (5, 155), (6, 20)] ... 180 0.7917 131317 [(0, 59438), (1, 47194), (2, 18633), (3, 4897), (4, 991), (5, 135), (6, 25)] ... 190 0.7862 130559 [(0, 59222), (1, 47118), (2, 18383), (3, 4769), (4, 906), (5, 136), (6, 23)] ... 200 0.7840 130547 [(0, 59355), (1, 46976), (2, 18455), (3, 4731), (4, 892), (5, 125), (6, 11)] ... 210 0.7843 130408 [(0, 59373), (1, 46835), (2, 18379), (3, 4775), (4, 891), (5, 140), (6, 13)] ... 220 0.7815 130078 [(0, 59291), (1, 46946), (2, 18112), (3, 4629), (4, 925), (5, 157), (6, 14)] ... 230 0.7762 129619 [(0, 59501), (1, 46465), (2, 18013), (3, 4625), (4, 862), (5, 128), (6, 23)] ... 240 0.7789 129309 [(0, 59004), (1, 46722), (2, 17930), (3, 4653), (4, 849), (5, 130), (6, 20)] ... 250 0.7773 129135 [(0, 59308), (1, 46218), (2, 17923), (3, 4635), (4, 881), (5, 141), (6, 28)] ... 260 0.7767 128965 [(0, 59323), (1, 46056), (2, 17875), (3, 4657), (4, 897), (5, 141), (6, 16)] ... 270 0.7749 128535 [(0, 59150), (1, 45992), (2, 17797), (3, 4557), (4, 870), (5, 150), (6, 17)] ... 280 0.7757 128331 [(0, 58838), (1, 46166), (2, 17808), (3, 4477), (4, 898), (5, 118), (6, 26)] ... 290 0.7699 128105 [(0, 59310), (1, 45712), (2, 17555), (3, 4497), (4, 866), (5, 147), (6, 12)] ... 300 0.7701 128335 [(0, 59404), (1, 45713), (2, 17694), (3, 4508), (4, 881), (5, 127), (6, 6)] ... 310 0.7651 127491 [(0, 59145), (1, 45604), (2, 17406), (3, 4377), (4, 823), (5, 113), (6, 21)] ... 320 0.7640 127538 [(0, 59433), (1, 45273), (2, 17445), (3, 4409), (4, 854), (5, 112), (6, 12)] ... 330 0.7628 127439 [(0, 59205), (1, 45650), (2, 17290), (3, 4325), (4, 858), (5, 93), (6, 18)] ... 340 0.7618 127120 [(0, 59100), (1, 45526), (2, 17248), (3, 4315), (4, 798), (5, 117), (6, 16)] ... 350 0.7651 126883 [(0, 59057), (1, 45099), (2, 17342), (3, 4409), (4, 823), (5, 142), (6, 10)] ... 360 0.7612 126766 [(0, 58994), (1, 45303), (2, 17287), (3, 4272), (4, 765), (5, 130), (6, 14)] ... 370 0.7594 126742 [(0, 59143), (1, 45280), (2, 17111), (3, 4250), (4, 820), (5, 121), (6, 14)] ... 380 0.7553 126453 [(0, 59379), (1, 44943), (2, 16907), (3, 4286), (4, 815), (5, 111), (6, 11)] ... 390 0.7562 126208 [(0, 59038), (1, 45136), (2, 16897), (3, 4178), (4, 828), (5, 118), (6, 13)] ... 400 0.7547 126191 [(0, 59185), (1, 44916), (2, 17051), (3, 4104), (4, 788), (5, 127), (6, 18)] ... 410 0.7570 125858 [(0, 58899), (1, 44756), (2, 17146), (3, 4145), (4, 783), (5, 113), (6, 15)] ... 420 0.7541 125957 [(0, 59114), (1, 44911), (2, 16787), (3, 4219), (4, 805), (5, 107), (6, 13)] ... 430 0.7500 125487 [(0, 59184), (1, 44500), (2, 16833), (3, 4078), (4, 769), (5, 100), (6, 21)] ... 440 0.7521 125564 [(0, 59019), (1, 44709), (2, 16845), (3, 4086), (4, 771), (5, 112), (6, 18)] ... 450 0.7459 125150 [(0, 59014), (1, 44817), (2, 16416), (3, 4042), (4, 741), (5, 105), (6, 14)] ... 460 0.7473 125371 [(0, 59248), (1, 44480), (2, 16733), (3, 4036), (4, 746), (5, 115), (6, 8)] ... 470 0.7489 125203 [(0, 58995), (1, 44609), (2, 16629), (3, 4137), (4, 706), (5, 103), (6, 20)] ... 480 0.7463 124951 [(0, 59136), (1, 44368), (2, 16494), (3, 4055), (4, 769), (5, 119), (6, 9)] ... 490 0.7480 124724 [(0, 58997), (1, 44164), (2, 16627), (3, 4017), (4, 789), (5, 113), (6, 14)] ... 500 0.7430 124838 [(0, 59217), (1, 44300), (2, 16481), (3, 3995), (4, 727), (5, 106), (6, 11)] ... 510 0.7480 124784 [(0, 58979), (1, 44228), (2, 16618), (3, 4094), (4, 749), (5, 106), (6, 7)] ... 520 0.7420 124577 [(0, 59264), (1, 44027), (2, 16425), (3, 4023), (4, 717), (5, 107), (6, 12)] ... 530 0.7400 124502 [(0, 59343), (1, 44031), (2, 16244), (3, 4040), (4, 741), (5, 86), (6, 16)] ... 540 0.7415 124261 [(0, 58981), (1, 44229), (2, 16226), (3, 3989), (4, 705), (5, 114), (6, 16)] ... 550 0.7389 124089 [(0, 58992), (1, 44144), (2, 16216), (3, 3942), (4, 695), (5, 89), (6, 11)] ... 560 0.7390 124210 [(0, 59092), (1, 44246), (2, 16053), (3, 3965), (4, 740), (5, 105), (6, 8)] ... 570 0.7398 124303 [(0, 59313), (1, 43905), (2, 16226), (3, 3968), (4, 775), (5, 96), (6, 20)] ... 580 0.7325 124037 [(0, 59505), (1, 43808), (2, 16035), (3, 3890), (4, 697), (5, 90), (6, 12)] ... 590 0.7367 123691 [(0, 59166), (1, 43698), (2, 15988), (3, 4014), (4, 726), (5, 86), (6, 12)] ... 600 0.7322 123609 [(0, 59364), (1, 43700), (2, 15809), (3, 3895), (4, 724), (5, 101), (6, 14)] ... 610 0.7364 123398 [(0, 59008), (1, 43674), (2, 15954), (3, 3881), (4, 773), (5, 90), (6, 17)] ... 620 0.7371 123247 [(0, 58967), (1, 43343), (2, 16200), (3, 3966), (4, 667), (5, 92), (6, 10)] ... 630 0.7354 123421 [(0, 58914), (1, 43795), (2, 16093), (3, 3812), (4, 697), (5, 100), (6, 9)] ... 640 0.7343 123140 [(0, 58930), (1, 43603), (2, 15955), (3, 3831), (4, 710), (5, 92), (6, 19)] ... 650 0.7345 123130 [(0, 59007), (1, 43480), (2, 15960), (3, 3835), (4, 719), (5, 120), (6, 9)] ... 660 0.7300 123003 [(0, 59247), (1, 43313), (2, 15832), (3, 3769), (4, 722), (5, 104), (6, 16)] ... 670 0.7357 122978 [(0, 58814), (1, 43432), (2, 16064), (3, 3856), (4, 719), (5, 83), (6, 10)] ... 680 0.7302 122872 [(0, 59237), (1, 43151), (2, 15798), (3, 3886), (4, 700), (5, 90), (6, 9)] ... 690 0.7289 122671 [(0, 59030), (1, 43338), (2, 15745), (3, 3779), (4, 664), (5, 96), (6, 17)] ... 700 0.7292 122669 [(0, 59087), (1, 43207), (2, 15810), (3, 3767), (4, 685), (5, 98), (6, 14)] ... 710 0.7249 122790 [(0, 59419), (1, 43197), (2, 15643), (3, 3735), (4, 679), (5, 101), (6, 13)] ... 720 0.7284 122322 [(0, 58939), (1, 43059), (2, 15837), (3, 3715), (4, 659), (5, 98), (6, 15)] ... 730 0.7306 122259 [(0, 58782), (1, 43058), (2, 15878), (3, 3786), (4, 636), (5, 106), (6, 12)] ... 740 0.7288 122166 [(0, 58921), (1, 43054), (2, 15530), (3, 3845), (4, 708), (5, 91), (6, 15)] ... 750 0.7257 122436 [(0, 59107), (1, 43179), (2, 15679), (3, 3687), (4, 676), (5, 98), (6, 9)] ... 760 0.7278 122255 [(0, 58838), (1, 43174), (2, 15812), (3, 3672), (4, 647), (5, 102), (6, 7)] ... 770 0.7226 122198 [(0, 59306), (1, 42800), (2, 15665), (3, 3650), (4, 675), (5, 89), (6, 12)] ... 780 0.7282 122212 [(0, 58899), (1, 42936), (2, 15885), (3, 3788), (4, 607), (5, 82), (6, 13)] ... 790 0.7232 121908 [(0, 58986), (1, 42970), (2, 15570), (3, 3607), (4, 666), (5, 95), (6, 9)] ... 800 0.7272 121835 [(0, 58772), (1, 42951), (2, 15601), (3, 3743), (4, 646), (5, 106), (6, 15)] ... 810 0.7236 121731 [(0, 58989), (1, 42776), (2, 15495), (3, 3704), (4, 648), (5, 107), (6, 10)] ... 820 0.7206 121914 [(0, 59216), (1, 42833), (2, 15428), (3, 3698), (4, 635), (5, 92), (6, 12)] ... 830 0.7219 121768 [(0, 59138), (1, 42682), (2, 15470), (3, 3725), (4, 662), (5, 82), (6, 8)] ... 840 0.7227 121701 [(0, 59035), (1, 42800), (2, 15345), (3, 3743), (4, 664), (5, 106), (6, 7)] ... 850 0.7211 121629 [(0, 58864), (1, 43123), (2, 15231), (3, 3663), (4, 628), (5, 105), (6, 14)] ... 860 0.7249 121311 [(0, 58665), (1, 42833), (2, 15260), (3, 3761), (4, 677), (5, 97), (6, 18)] ... 870 0.7199 121557 [(0, 59173), (1, 42477), (2, 15560), (3, 3601), (4, 631), (5, 105), (6, 10)] ... 880 0.7188 121021 [(0, 58833), (1, 42665), (2, 15143), (3, 3612), (4, 656), (5, 97), (6, 13)] ... 890 0.7189 121088 [(0, 58788), (1, 42809), (2, 15110), (3, 3613), (4, 660), (5, 101), (6, 7)] ... 900 0.7157 121521 [(0, 59371), (1, 42411), (2, 15458), (3, 3570), (4, 621), (5, 84), (6, 6)] ... 910 0.7158 121371 [(0, 59209), (1, 42603), (2, 15276), (3, 3533), (4, 643), (5, 89), (6, 16)] ... 920 0.7218 121064 [(0, 58832), (1, 42394), (2, 15399), (3, 3682), (4, 648), (5, 98), (6, 9)] ... 930 0.7207 120924 [(0, 58765), (1, 42438), (2, 15316), (3, 3648), (4, 668), (5, 78), (6, 8)] ... 940 0.7212 121278 [(0, 58845), (1, 42713), (2, 15315), (3, 3634), (4, 654), (5, 100), (6, 16)] ... 950 0.7164 120639 [(0, 58779), (1, 42319), (2, 15327), (3, 3518), (4, 595), (5, 90), (6, 10)] ... 960 0.7135 121021 [(0, 59303), (1, 42218), (2, 15178), (3, 3613), (4, 618), (5, 81), (6, 10)] ... 970 0.7181 120517 [(0, 58726), (1, 42251), (2, 15213), (3, 3553), (4, 673), (5, 91), (6, 10)] ... 980 0.7122 120989 [(0, 59264), (1, 42355), (2, 15091), (3, 3597), (4, 587), (5, 85), (6, 8)] ... 990 0.7106 120879 [(0, 59390), (1, 42202), (2, 15003), (3, 3546), (4, 643), (5, 88), (6, 7)] ... ******** For exponents with p = 3 (mod 8) 000 1.0603 166161 [(0, 57217), (1, 61404), (2, 32452), (3, 11366), (4, 3001), (5, 583), (6, 119)] ... 010 0.9707 151607 [(0, 57332), (1, 55810), (2, 27140), (3, 8758), (4, 2112), (5, 391), (6, 54)] ... 020 0.9365 146781 [(0, 57306), (1, 54065), (2, 25498), (3, 7708), (4, 1818), (5, 327), (6, 50)] ... 030 0.9209 143964 [(0, 56971), (1, 53256), (2, 24339), (3, 7412), (4, 1605), (5, 314), (6, 58)] ... 040 0.9112 141848 [(0, 56711), (1, 52213), (2, 24018), (3, 7013), (4, 1542), (5, 310), (6, 36)] ... 050 0.8941 140198 [(0, 57012), (1, 51552), (2, 23170), (3, 6712), (4, 1468), (5, 254), (6, 26)] ... 060 0.8845 139055 [(0, 57260), (1, 50934), (2, 22566), (3, 6559), (4, 1461), (5, 240), (6, 31)] ... 070 0.8777 137940 [(0, 56986), (1, 50754), (2, 22160), (3, 6452), (4, 1333), (5, 224), (6, 27)] ... 080 0.8696 136764 [(0, 57139), (1, 50065), (2, 21726), (3, 6192), (4, 1406), (5, 202), (6, 29)] ... 090 0.8642 136226 [(0, 57171), (1, 49903), (2, 21440), (3, 6176), (4, 1299), (5, 207), (6, 25)] ... 100 0.8583 135539 [(0, 57195), (1, 49717), (2, 21060), (3, 6053), (4, 1271), (5, 213), (6, 25)] ... 110 0.8596 134412 [(0, 56727), (1, 49188), (2, 20961), (3, 6009), (4, 1272), (5, 222), (6, 26)] ... 120 0.8509 134158 [(0, 57260), (1, 48734), (2, 20825), (3, 5872), (4, 1229), (5, 197), (6, 38)] ... 130 0.8486 133367 [(0, 56664), (1, 48916), (2, 20774), (3, 5631), (4, 1143), (5, 192), (6, 40)] ... 140 0.8390 133140 [(0, 57313), (1, 48493), (2, 20397), (3, 5576), (4, 1146), (5, 183), (6, 28)] ... 150 0.8373 132587 [(0, 56993), (1, 48687), (2, 19974), (3, 5585), (4, 1144), (5, 174), (6, 26)] ... 160 0.8351 132425 [(0, 57016), (1, 48541), (2, 20129), (3, 5417), (4, 1101), (5, 191), (6, 28)] ... 170 0.8319 131739 [(0, 57176), (1, 47912), (2, 19831), (3, 5489), (4, 1148), (5, 147), (6, 32)] ... 180 0.8316 131245 [(0, 56816), (1, 47931), (2, 19794), (3, 5410), (4, 1095), (5, 178), (6, 20)] ... 190 0.8306 130950 [(0, 56897), (1, 47443), (2, 20037), (3, 5272), (4, 1088), (5, 195), (6, 15)] ... 200 0.8259 130795 [(0, 57107), (1, 47427), (2, 19643), (3, 5355), (4, 1087), (5, 157), (6, 17)] ... 210 0.8226 129932 [(0, 56798), (1, 47252), (2, 19409), (3, 5266), (4, 1034), (5, 157), (6, 15)] ... 220 0.8202 130148 [(0, 57049), (1, 47237), (2, 19427), (3, 5278), (4, 981), (5, 158), (6, 18)] ... 230 0.8184 129799 [(0, 57005), (1, 47073), (2, 19427), (3, 5074), (4, 1044), (5, 158), (6, 15)] ... 240 0.8165 129623 [(0, 57092), (1, 46994), (2, 19224), (3, 5079), (4, 1041), (5, 168), (6, 21)] ... 250 0.8180 128934 [(0, 56498), (1, 47121), (2, 19038), (3, 5084), (4, 982), (5, 182), (6, 23)] ... 260 0.8148 128479 [(0, 56650), (1, 46698), (2, 18784), (3, 5153), (4, 1033), (5, 139), (6, 21)] ... 270 0.8105 128758 [(0, 56855), (1, 46999), (2, 18749), (3, 4962), (4, 1020), (5, 148), (6, 22)] ... 280 0.8061 128459 [(0, 57089), (1, 46698), (2, 18491), (3, 5050), (4, 956), (5, 151), (6, 21)] ... 290 0.8069 128119 [(0, 56784), (1, 46547), (2, 18859), (3, 4807), (4, 938), (5, 160), (6, 23)] ... 300 0.8069 127922 [(0, 56993), (1, 46122), (2, 18696), (3, 4934), (4, 1009), (5, 144), (6, 20)] ... 310 0.8068 127706 [(0, 56743), (1, 46327), (2, 18565), (3, 4930), (4, 946), (5, 173), (6, 17)] ... 320 0.8041 127674 [(0, 56841), (1, 46402), (2, 18372), (3, 4908), (4, 984), (5, 145), (6, 20)] ... 330 0.8003 127195 [(0, 56682), (1, 46261), (2, 18448), (3, 4737), (4, 924), (5, 130), (6, 11)] ... 340 0.7978 127514 [(0, 57264), (1, 45963), (2, 18397), (3, 4777), (4, 945), (5, 146), (6, 18)] ... 350 0.7950 126936 [(0, 57097), (1, 45920), (2, 18068), (3, 4744), (4, 928), (5, 162), (6, 15)] ... 360 0.7968 126590 [(0, 56784), (1, 45816), (2, 18140), (3, 4797), (4, 907), (5, 129), (6, 15)] ... 370 0.7992 126963 [(0, 56762), (1, 45985), (2, 18410), (3, 4738), (4, 917), (5, 127), (6, 22)] ... 380 0.7953 126283 [(0, 56681), (1, 45756), (2, 18094), (3, 4688), (4, 911), (5, 138), (6, 13)] ... 390 0.7968 126318 [(0, 56768), (1, 45526), (2, 18206), (3, 4764), (4, 884), (5, 140), (6, 29)] ... 400 0.7934 125919 [(0, 56790), (1, 45394), (2, 17954), (3, 4704), (4, 910), (5, 149), (6, 16)] ... 410 0.7880 126062 [(0, 57107), (1, 45398), (2, 17982), (3, 4528), (4, 872), (5, 153), (6, 19)] ... 420 0.7863 125863 [(0, 57064), (1, 45412), (2, 17774), (3, 4617), (4, 841), (5, 140), (6, 14)] ... 430 0.7838 125847 [(0, 57348), (1, 45211), (2, 17667), (3, 4568), (4, 892), (5, 140), (6, 21)] ... 440 0.7876 125771 [(0, 56749), (1, 45655), (2, 17858), (3, 4512), (4, 855), (5, 126), (6, 14)] ... 450 0.7805 125369 [(0, 57340), (1, 44898), (2, 17620), (3, 4513), (4, 841), (5, 139), (6, 16)] ... 460 0.7848 125306 [(0, 57021), (1, 44932), (2, 17820), (3, 4524), (4, 873), (5, 113), (6, 20)] ... 470 0.7885 125163 [(0, 56555), (1, 45336), (2, 17680), (3, 4538), (4, 909), (5, 127), (6, 14)] ... 480 0.7823 124870 [(0, 56713), (1, 45268), (2, 17437), (3, 4453), (4, 840), (5, 136), (6, 21)] ... 490 0.7820 125005 [(0, 56927), (1, 45036), (2, 17599), (3, 4433), (4, 843), (5, 148), (6, 19)] ... 500 0.7777 124475 [(0, 57001), (1, 44698), (2, 17448), (3, 4271), (4, 905), (5, 133), (6, 17)] ... 510 0.7801 124622 [(0, 56808), (1, 44941), (2, 17510), (3, 4357), (4, 865), (5, 125), (6, 15)] ... 520 0.7808 124536 [(0, 56808), (1, 44724), (2, 17620), (3, 4411), (4, 842), (5, 112), (6, 18)] ... 530 0.7815 124366 [(0, 56627), (1, 44957), (2, 17268), (3, 4507), (4, 868), (5, 124), (6, 14)] ... 540 0.7790 124214 [(0, 56736), (1, 44738), (2, 17385), (3, 4327), (4, 887), (5, 123), (6, 18)] ... 550 0.7775 124204 [(0, 56687), (1, 44894), (2, 17290), (3, 4384), (4, 815), (5, 121), (6, 12)] ... 560 0.7732 123865 [(0, 57088), (1, 44210), (2, 17225), (3, 4400), (4, 816), (5, 110), (6, 16)] ... 570 0.7789 123782 [(0, 56724), (1, 44292), (2, 17340), (3, 4433), (4, 854), (5, 114), (6, 20)] ... 580 0.7761 123930 [(0, 56945), (1, 44324), (2, 17225), (3, 4471), (4, 839), (5, 114), (6, 11)] ... 590 0.7719 123844 [(0, 57041), (1, 44411), (2, 17110), (3, 4312), (4, 839), (5, 113), (6, 16)] ... 600 0.7696 123806 [(0, 57030), (1, 44539), (2, 17065), (3, 4245), (4, 788), (5, 114), (6, 24)] ... 610 0.7680 123709 [(0, 57121), (1, 44484), (2, 16900), (3, 4262), (4, 797), (5, 124), (6, 19)] ... 620 0.7700 123412 [(0, 56956), (1, 44157), (2, 17118), (3, 4244), (4, 797), (5, 120), (6, 20)] ... 630 0.7719 123358 [(0, 56832), (1, 44263), (2, 17008), (3, 4251), (4, 854), (5, 132), (6, 16)] ... 640 0.7712 123199 [(0, 56599), (1, 44517), (2, 16899), (3, 4218), (4, 814), (5, 129), (6, 19)] ... 650 0.7708 123306 [(0, 56857), (1, 44138), (2, 17081), (3, 4329), (4, 761), (5, 122), (6, 18)] ... 660 0.7689 122789 [(0, 56723), (1, 43987), (2, 16922), (3, 4209), (4, 808), (5, 121), (6, 14)] ... 670 0.7643 122987 [(0, 57017), (1, 44118), (2, 16746), (3, 4183), (4, 796), (5, 106), (6, 19)] ... 680 0.7664 123042 [(0, 56849), (1, 44245), (2, 16825), (3, 4228), (4, 763), (5, 118), (6, 14)] ... 690 0.7656 122854 [(0, 56852), (1, 44046), (2, 16874), (3, 4201), (4, 760), (5, 107), (6, 13)] ... 700 0.7671 122606 [(0, 56828), (1, 43714), (2, 16898), (3, 4272), (4, 762), (5, 116), (6, 15)] ... 710 0.7644 122472 [(0, 56645), (1, 44087), (2, 16687), (3, 4208), (4, 719), (5, 108), (6, 16)] ... 720 0.7687 122701 [(0, 56737), (1, 43975), (2, 16768), (3, 4249), (4, 828), (5, 122), (6, 19)] ... 730 0.7634 122609 [(0, 56756), (1, 44132), (2, 16724), (3, 4109), (4, 755), (5, 121), (6, 11)] ... 740 0.7572 122391 [(0, 57098), (1, 43826), (2, 16578), (3, 4010), (4, 753), (5, 110), (6, 15)] ... 750 0.7598 122224 [(0, 56918), (1, 43659), (2, 16732), (3, 4075), (4, 701), (5, 122), (6, 17)] ... 760 0.7611 121824 [(0, 56712), (1, 43467), (2, 16715), (3, 4013), (4, 812), (5, 89), (6, 15)] ... 770 0.7634 122009 [(0, 56696), (1, 43597), (2, 16668), (3, 4148), (4, 750), (5, 133), (6, 14)] ... 780 0.7607 122205 [(0, 57026), (1, 43530), (2, 16602), (3, 4122), (4, 785), (5, 125), (6, 14)] ... 790 0.7590 122086 [(0, 56975), (1, 43616), (2, 16518), (3, 4059), (4, 781), (5, 118), (6, 18)] ... 800 0.7617 122069 [(0, 56774), (1, 43633), (2, 16652), (3, 4121), (4, 783), (5, 95), (6, 10)] ... 810 0.7590 121864 [(0, 56744), (1, 43738), (2, 16400), (3, 4095), (4, 779), (5, 99), (6, 8)] ... 820 0.7601 121540 [(0, 56582), (1, 43495), (2, 16576), (3, 3959), (4, 798), (5, 116), (6, 14)] ... 830 0.7542 121998 [(0, 57217), (1, 43466), (2, 16426), (3, 4031), (4, 716), (5, 125), (6, 14)] ... 840 0.7562 121675 [(0, 56886), (1, 43516), (2, 16303), (3, 4118), (4, 742), (5, 98), (6, 11)] ... 850 0.7582 121201 [(0, 56482), (1, 43330), (2, 16560), (3, 4003), (4, 716), (5, 95), (6, 14)] ... 860 0.7538 121813 [(0, 57151), (1, 43448), (2, 16298), (3, 4028), (4, 762), (5, 115), (6, 10)] ... 870 0.7528 121328 [(0, 56871), (1, 43313), (2, 16347), (3, 3975), (4, 712), (5, 99), (6, 11)] ... 880 0.7536 121358 [(0, 56858), (1, 43336), (2, 16340), (3, 3979), (4, 737), (5, 100), (6, 7)] ... 890 0.7572 121121 [(0, 56517), (1, 43366), (2, 16340), (3, 4040), (4, 753), (5, 93), (6, 10)] ... 900 0.7580 121131 [(0, 56523), (1, 43391), (2, 16259), (3, 4076), (4, 743), (5, 129), (6, 8)] ... 910 0.7490 121251 [(0, 57079), (1, 43244), (2, 16110), (3, 4037), (4, 675), (5, 93), (6, 12)] ... 920 0.7516 121036 [(0, 56824), (1, 43249), (2, 16138), (3, 3986), (4, 722), (5, 106), (6, 10)] ... 930 0.7493 120894 [(0, 57036), (1, 42915), (2, 16121), (3, 3981), (4, 723), (5, 114), (6, 3)] ... 940 0.7522 120899 [(0, 56670), (1, 43392), (2, 15980), (3, 3985), (4, 743), (5, 115), (6, 13)] ... 950 0.7482 121400 [(0, 57262), (1, 43303), (2, 15974), (3, 3976), (4, 777), (5, 101), (6, 7)] ... 960 0.7498 120769 [(0, 56811), (1, 43073), (2, 16147), (3, 3903), (4, 722), (5, 96), (6, 15)] ... 970 0.7475 120726 [(0, 57066), (1, 42910), (2, 15907), (3, 3985), (4, 741), (5, 108), (6, 9)] ... 980 0.7482 120611 [(0, 56749), (1, 43185), (2, 15978), (3, 3841), (4, 731), (5, 115), (6, 11)] ... 990 0.7484 120801 [(0, 56848), (1, 43188), (2, 16050), (3, 3899), (4, 695), (5, 94), (6, 22)] ... ******** For exponents with p = 5 (mod 8) 000 1.0131 166204 [(0, 60377), (1, 61158), (2, 30874), (3, 10504), (4, 2628), (5, 550), (6, 94)] ... 010 0.9218 151489 [(0, 59886), (1, 55999), (2, 25705), (3, 7771), (4, 1770), (5, 314), (6, 35)] ... 020 0.8977 146798 [(0, 59652), (1, 53862), (2, 24244), (3, 7100), (4, 1609), (5, 292), (6, 34)] ... 030 0.8763 144096 [(0, 59798), (1, 52917), (2, 22851), (3, 6797), (4, 1446), (5, 251), (6, 32)] ... 040 0.8609 141846 [(0, 59936), (1, 51651), (2, 22245), (3, 6368), (4, 1393), (5, 216), (6, 35)] ... 050 0.8552 140271 [(0, 59331), (1, 51341), (2, 21993), (3, 6066), (4, 1307), (5, 202), (6, 26)] ... 060 0.8475 139032 [(0, 59358), (1, 50828), (2, 21323), (3, 5991), (4, 1304), (5, 198), (6, 29)] ... 070 0.8347 137838 [(0, 59628), (1, 50174), (2, 20873), (3, 5776), (4, 1159), (5, 206), (6, 21)] ... 080 0.8270 137034 [(0, 59827), (1, 49675), (2, 20520), (3, 5685), (4, 1103), (5, 197), (6, 24)] ... 090 0.8230 135926 [(0, 59518), (1, 49382), (2, 20153), (3, 5556), (4, 1104), (5, 182), (6, 27)] ... 100 0.8182 135563 [(0, 59745), (1, 48959), (2, 20100), (3, 5490), (4, 1082), (5, 167), (6, 17)] ... 110 0.8150 134455 [(0, 59271), (1, 48844), (2, 19753), (3, 5365), (4, 1009), (5, 177), (6, 32)] ... 120 0.8150 134037 [(0, 59073), (1, 48681), (2, 19722), (3, 5322), (4, 1071), (5, 145), (6, 19)] ... 130 0.8100 133644 [(0, 59243), (1, 48447), (2, 19522), (3, 5196), (4, 1041), (5, 163), (6, 25)] ... 140 0.8029 133164 [(0, 59466), (1, 48073), (2, 19383), (3, 5084), (4, 995), (5, 141), (6, 18)] ... 150 0.8006 132157 [(0, 59089), (1, 47913), (2, 18965), (3, 5004), (4, 998), (5, 168), (6, 16)] ... 160 0.7970 132144 [(0, 59340), (1, 47703), (2, 18961), (3, 5045), (4, 938), (5, 139), (6, 16)] ... 170 0.7979 131849 [(0, 59154), (1, 47631), (2, 18950), (3, 4986), (4, 961), (5, 141), (6, 23)] ... 180 0.7981 131347 [(0, 59044), (1, 47303), (2, 18861), (3, 4961), (4, 995), (5, 166), (6, 15)] ... 190 0.7940 130903 [(0, 59125), (1, 47108), (2, 18591), (3, 4895), (4, 992), (5, 165), (6, 25)] ... 200 0.7845 130721 [(0, 59590), (1, 46807), (2, 18498), (3, 4749), (4, 910), (5, 149), (6, 15)] ... 210 0.7894 130355 [(0, 59053), (1, 46913), (2, 18432), (3, 4886), (4, 915), (5, 131), (6, 24)] ... 220 0.7842 129853 [(0, 59274), (1, 46422), (2, 18361), (3, 4682), (4, 953), (5, 140), (6, 19)] ... 230 0.7800 129858 [(0, 59250), (1, 46852), (2, 18078), (3, 4615), (4, 898), (5, 147), (6, 14)] ... 240 0.7787 129406 [(0, 59345), (1, 46275), (2, 18099), (3, 4655), (4, 856), (5, 154), (6, 22)] ... 250 0.7786 129268 [(0, 59153), (1, 46431), (2, 18040), (3, 4615), (4, 876), (5, 129), (6, 20)] ... 260 0.7754 129161 [(0, 59244), (1, 46449), (2, 17903), (3, 4563), (4, 832), (5, 145), (6, 21)] ... 270 0.7754 128709 [(0, 59311), (1, 45854), (2, 17888), (3, 4622), (4, 893), (5, 120), (6, 16)] ... 280 0.7760 128399 [(0, 59064), (1, 45862), (2, 17794), (3, 4690), (4, 844), (5, 129), (6, 15)] ... 290 0.7700 128136 [(0, 59300), (1, 45803), (2, 17499), (3, 4457), (4, 908), (5, 147), (6, 22)] ... 300 0.7712 127953 [(0, 59093), (1, 45652), (2, 17771), (3, 4432), (4, 853), (5, 138), (6, 11)] ... 310 0.7728 127903 [(0, 58957), (1, 45804), (2, 17569), (3, 4576), (4, 837), (5, 138), (6, 19)] ... 320 0.7687 127390 [(0, 58974), (1, 45613), (2, 17290), (3, 4487), (4, 884), (5, 122), (6, 20)] ... 330 0.7636 127244 [(0, 59142), (1, 45513), (2, 17246), (3, 4367), (4, 832), (5, 132), (6, 11)] ... 340 0.7608 127041 [(0, 59265), (1, 45340), (2, 17069), (3, 4425), (4, 836), (5, 84), (6, 18)] ... 350 0.7661 126872 [(0, 58798), (1, 45551), (2, 17082), (3, 4438), (4, 868), (5, 119), (6, 14)] ... 360 0.7585 126944 [(0, 59360), (1, 45225), (2, 17060), (3, 4384), (4, 791), (5, 114), (6, 10)] ... 370 0.7587 126661 [(0, 59202), (1, 45098), (2, 17130), (3, 4326), (4, 779), (5, 111), (6, 14)] ... 380 0.7565 126624 [(0, 59260), (1, 45232), (2, 16928), (3, 4273), (4, 786), (5, 130), (6, 15)] ... 390 0.7579 126282 [(0, 59085), (1, 45039), (2, 16945), (3, 4256), (4, 796), (5, 146), (6, 12)] ... 400 0.7537 126064 [(0, 59329), (1, 44508), (2, 17145), (3, 4235), (4, 739), (5, 100), (6, 6)] ... 410 0.7513 126066 [(0, 59375), (1, 44871), (2, 16702), (3, 4191), (4, 789), (5, 120), (6, 17)] ... 420 0.7547 125633 [(0, 58912), (1, 44737), (2, 16945), (3, 4122), (4, 773), (5, 132), (6, 12)] ... 430 0.7497 125538 [(0, 59177), (1, 44518), (2, 16898), (3, 4109), (4, 725), (5, 94), (6, 16)] ... 440 0.7505 125971 [(0, 59386), (1, 44752), (2, 16784), (3, 4121), (4, 797), (5, 115), (6, 14)] ... 450 0.7493 125476 [(0, 59186), (1, 44564), (2, 16770), (3, 4064), (4, 755), (5, 125), (6, 8)] ... 460 0.7537 125058 [(0, 58590), (1, 44652), (2, 16872), (3, 4057), (4, 764), (5, 109), (6, 13)] ... 470 0.7450 125125 [(0, 59198), (1, 44575), (2, 16400), (3, 4110), (4, 721), (5, 104), (6, 15)] ... 480 0.7467 124947 [(0, 59155), (1, 44240), (2, 16574), (3, 4139), (4, 720), (5, 102), (6, 16)] ... 490 0.7445 124867 [(0, 59184), (1, 44372), (2, 16337), (3, 4104), (4, 749), (5, 114), (6, 6)] ... 500 0.7403 124575 [(0, 59293), (1, 44127), (2, 16314), (3, 4028), (4, 698), (5, 104), (6, 10)] ... 510 0.7481 124496 [(0, 58821), (1, 44163), (2, 16541), (3, 4119), (4, 745), (5, 96), (6, 9)] ... 520 0.7416 124488 [(0, 59083), (1, 44273), (2, 16286), (3, 4048), (4, 678), (5, 105), (6, 14)] ... 530 0.7471 124395 [(0, 58815), (1, 44071), (2, 16641), (3, 4011), (4, 750), (5, 96), (6, 9)] ... 540 0.7359 124319 [(0, 59383), (1, 44090), (2, 16107), (3, 3911), (4, 712), (5, 101), (6, 12)] ... 550 0.7384 124252 [(0, 59343), (1, 43847), (2, 16258), (3, 3961), (4, 732), (5, 98), (6, 11)] ... 560 0.7378 123752 [(0, 59088), (1, 43785), (2, 16083), (3, 3959), (4, 723), (5, 96), (6, 18)] ... 570 0.7386 124013 [(0, 59118), (1, 43935), (2, 16177), (3, 3938), (4, 747), (5, 86), (6, 10)] ... 580 0.7374 123626 [(0, 59193), (1, 43515), (2, 16075), (3, 4008), (4, 711), (5, 113), (6, 10)] ... 590 0.7392 123914 [(0, 58994), (1, 43972), (2, 16175), (3, 3939), (4, 723), (5, 101), (6, 9)] ... 600 0.7371 123640 [(0, 59112), (1, 43617), (2, 16151), (3, 3948), (4, 706), (5, 93), (6, 13)] ... 610 0.7381 123327 [(0, 58907), (1, 43443), (2, 16300), (3, 3843), (4, 730), (5, 89), (6, 15)] ... 620 0.7325 123626 [(0, 59257), (1, 43792), (2, 15893), (3, 3869), (4, 709), (5, 97), (6, 7)] ... 630 0.7383 123188 [(0, 58770), (1, 43553), (2, 16176), (3, 3831), (4, 747), (5, 103), (6, 7)] ... 640 0.7331 123468 [(0, 59174), (1, 43596), (2, 16103), (3, 3796), (4, 690), (5, 94), (6, 14)] ... 650 0.7287 122908 [(0, 59035), (1, 43692), (2, 15593), (3, 3789), (4, 682), (5, 112), (6, 5)] ... 660 0.7348 123116 [(0, 59042), (1, 43363), (2, 16035), (3, 3814), (4, 739), (5, 102), (6, 21)] ... 670 0.7352 123113 [(0, 58837), (1, 43704), (2, 15899), (3, 3822), (4, 724), (5, 116), (6, 10)] ... 680 0.7279 122921 [(0, 59089), (1, 43643), (2, 15607), (3, 3822), (4, 656), (5, 98), (6, 6)] ... 690 0.7292 122810 [(0, 59125), (1, 43345), (2, 15731), (3, 3794), (4, 721), (5, 85), (6, 9)] ... 700 0.7307 122953 [(0, 59139), (1, 43377), (2, 15799), (3, 3818), (4, 704), (5, 106), (6, 8)] ... 710 0.7284 122381 [(0, 58813), (1, 43305), (2, 15810), (3, 3712), (4, 637), (5, 93), (6, 11)] ... 720 0.7249 122616 [(0, 59267), (1, 43220), (2, 15622), (3, 3731), (4, 658), (5, 107), (6, 11)] ... 730 0.7286 122479 [(0, 59051), (1, 43148), (2, 15639), (3, 3857), (4, 687), (5, 89), (6, 7)] ... 740 0.7251 122230 [(0, 58971), (1, 43178), (2, 15664), (3, 3659), (4, 651), (5, 100), (6, 7)] ... 750 0.7273 122269 [(0, 58932), (1, 43209), (2, 15586), (3, 3767), (4, 652), (5, 108), (6, 14)] ... 760 0.7234 122552 [(0, 59180), (1, 43326), (2, 15706), (3, 3555), (4, 682), (5, 89), (6, 14)] ... 770 0.7247 121934 [(0, 59026), (1, 42871), (2, 15549), (3, 3693), (4, 669), (5, 111), (6, 14)] ... 780 0.7265 122248 [(0, 58973), (1, 43101), (2, 15719), (3, 3681), (4, 665), (5, 89), (6, 19)] ... 790 0.7279 122067 [(0, 58597), (1, 43405), (2, 15601), (3, 3730), (4, 628), (5, 94), (6, 10)] ... 800 0.7236 122262 [(0, 59260), (1, 42982), (2, 15447), (3, 3822), (4, 647), (5, 89), (6, 14)] ... 810 0.7253 121805 [(0, 58924), (1, 42787), (2, 15636), (3, 3673), (4, 669), (5, 106), (6, 10)] ... 820 0.7197 121742 [(0, 59086), (1, 42867), (2, 15405), (3, 3685), (4, 615), (5, 78), (6, 6)] ... 830 0.7221 121718 [(0, 58882), (1, 43085), (2, 15335), (3, 3647), (4, 660), (5, 93), (6, 16)] ... 840 0.7239 121380 [(0, 58818), (1, 42618), (2, 15470), (3, 3703), (4, 665), (5, 97), (6, 8)] ... 850 0.7203 121408 [(0, 58903), (1, 42834), (2, 15295), (3, 3606), (4, 652), (5, 104), (6, 12)] ... 860 0.7238 121471 [(0, 58780), (1, 42757), (2, 15523), (3, 3661), (4, 633), (5, 104), (6, 13)] ... 870 0.7184 121381 [(0, 59065), (1, 42682), (2, 15235), (3, 3671), (4, 624), (5, 87), (6, 15)] ... 880 0.7186 121329 [(0, 59055), (1, 42581), (2, 15296), (3, 3677), (4, 629), (5, 78), (6, 11)] ... 890 0.7220 121448 [(0, 58815), (1, 42909), (2, 15310), (3, 3630), (4, 675), (5, 95), (6, 12)] ... 900 0.7132 121389 [(0, 59502), (1, 42380), (2, 15161), (3, 3625), (4, 620), (5, 87), (6, 12)] ... 910 0.7151 121253 [(0, 59225), (1, 42564), (2, 15078), (3, 3667), (4, 623), (5, 83), (6, 11)] ... 920 0.7192 120749 [(0, 58793), (1, 42331), (2, 15255), (3, 3580), (4, 693), (5, 92), (6, 5)] ... 930 0.7214 121311 [(0, 58975), (1, 42537), (2, 15293), (3, 3770), (4, 613), (5, 115), (6, 7)] ... 940 0.7169 120899 [(0, 58936), (1, 42474), (2, 15112), (3, 3636), (4, 651), (5, 75), (6, 15)] ... 950 0.7186 121063 [(0, 58765), (1, 42720), (2, 15314), (3, 3512), (4, 663), (5, 81), (6, 6)] ... 960 0.7150 120829 [(0, 58821), (1, 42697), (2, 15043), (3, 3554), (4, 635), (5, 70), (6, 9)] ... 970 0.7124 120854 [(0, 59337), (1, 42086), (2, 15100), (3, 3618), (4, 619), (5, 81), (6, 13)] ... 980 0.7137 120572 [(0, 58976), (1, 42300), (2, 14993), (3, 3562), (4, 639), (5, 93), (6, 8)] ... 990 0.7168 120410 [(0, 58740), (1, 42234), (2, 15073), (3, 3633), (4, 628), (5, 89), (6, 12)] ... ******** For exponents with p = 7 (mod 8) 000 1.0595 166237 [(0, 57289), (1, 61523), (2, 32301), (3, 11390), (4, 2992), (5, 603), (6, 120)] ... 010 0.9676 151431 [(0, 57386), (1, 55767), (2, 27085), (3, 8710), (4, 2040), (5, 374), (6, 62)] ... 020 0.9351 146862 [(0, 57305), (1, 54299), (2, 25323), (3, 7792), (4, 1772), (5, 318), (6, 41)] ... 030 0.9166 143923 [(0, 57229), (1, 53150), (2, 24282), (3, 7252), (4, 1664), (5, 289), (6, 50)] ... 040 0.9057 141867 [(0, 57062), (1, 52324), (2, 23560), (3, 7019), (4, 1572), (5, 282), (6, 45)] ... 050 0.8943 140565 [(0, 57106), (1, 51748), (2, 23275), (3, 6671), (4, 1465), (5, 262), (6, 34)] ... 060 0.8868 138846 [(0, 56803), (1, 51257), (2, 22514), (3, 6588), (4, 1377), (5, 271), (6, 31)] ... 070 0.8784 137723 [(0, 56937), (1, 50471), (2, 22334), (3, 6373), (4, 1350), (5, 228), (6, 27)] ... 080 0.8723 136788 [(0, 56793), (1, 50338), (2, 21843), (3, 6225), (4, 1346), (5, 219), (6, 22)] ... 090 0.8656 136164 [(0, 57171), (1, 49754), (2, 21455), (3, 6246), (4, 1273), (5, 229), (6, 33)] ... 100 0.8587 135461 [(0, 57019), (1, 49810), (2, 21191), (3, 5929), (4, 1248), (5, 234), (6, 28)] ... 110 0.8530 134810 [(0, 57055), (1, 49537), (2, 20867), (3, 5945), (4, 1179), (5, 189), (6, 35)] ... 120 0.8510 133992 [(0, 56930), (1, 49014), (2, 20872), (3, 5716), (4, 1219), (5, 212), (6, 24)] ... 130 0.8481 133589 [(0, 56954), (1, 48784), (2, 20680), (3, 5760), (4, 1207), (5, 182), (6, 21)] ... 140 0.8476 132993 [(0, 56638), (1, 48695), (2, 20564), (3, 5734), (4, 1131), (5, 213), (6, 18)] ... 150 0.8382 132736 [(0, 57055), (1, 48601), (2, 20193), (3, 5513), (4, 1160), (5, 188), (6, 23)] ... 160 0.8375 131974 [(0, 56681), (1, 48442), (2, 19996), (3, 5565), (4, 1075), (5, 194), (6, 18)] ... 170 0.8333 131818 [(0, 57119), (1, 47864), (2, 20079), (3, 5435), (4, 1118), (5, 179), (6, 20)] ... 180 0.8323 131179 [(0, 56786), (1, 47803), (2, 19949), (3, 5343), (4, 1073), (5, 200), (6, 20)] ... 190 0.8313 131052 [(0, 56682), (1, 47974), (2, 19786), (3, 5280), (4, 1120), (5, 190), (6, 19)] ... 200 0.8243 130626 [(0, 56795), (1, 47879), (2, 19495), (3, 5210), (4, 1083), (5, 137), (6, 24)] ... 210 0.8242 130219 [(0, 56991), (1, 47110), (2, 19599), (3, 5281), (4, 1031), (5, 187), (6, 18)] ... 220 0.8173 129954 [(0, 57240), (1, 47044), (2, 19260), (3, 5196), (4, 1031), (5, 163), (6, 17)] ... 230 0.8205 129695 [(0, 56910), (1, 47033), (2, 19355), (3, 5130), (4, 1082), (5, 164), (6, 19)] ... 240 0.8138 129436 [(0, 56992), (1, 47125), (2, 19134), (3, 4978), (4, 1047), (5, 139), (6, 18)] ... 250 0.8130 129205 [(0, 56982), (1, 46987), (2, 19020), (3, 5030), (4, 1020), (5, 146), (6, 19)] ... 260 0.8138 128919 [(0, 56935), (1, 46766), (2, 18915), (3, 5095), (4, 1026), (5, 159), (6, 22)] ... 270 0.8055 128683 [(0, 57149), (1, 46772), (2, 18757), (3, 4861), (4, 965), (5, 157), (6, 18)] ... 280 0.8111 128405 [(0, 56780), (1, 46616), (2, 18867), (3, 4987), (4, 976), (5, 148), (6, 27)] ... 290 0.8096 128302 [(0, 56806), (1, 46542), (2, 18904), (3, 4895), (4, 969), (5, 164), (6, 18)] ... 300 0.8050 127986 [(0, 57037), (1, 46238), (2, 18631), (3, 4983), (4, 923), (5, 151), (6, 23)] ... 310 0.8050 127585 [(0, 56960), (1, 45921), (2, 18630), (3, 4960), (4, 946), (5, 146), (6, 19)] ... 320 0.8020 127667 [(0, 57074), (1, 46094), (2, 18488), (3, 4911), (4, 932), (5, 145), (6, 22)] ... 330 0.7999 127252 [(0, 56762), (1, 46240), (2, 18475), (3, 4680), (4, 933), (5, 143), (6, 17)] ... 340 0.8020 127246 [(0, 56981), (1, 45843), (2, 18373), (3, 4924), (4, 944), (5, 169), (6, 11)] ... 350 0.8018 127071 [(0, 56498), (1, 46347), (2, 18435), (3, 4696), (4, 916), (5, 161), (6, 14)] ... 360 0.7961 126817 [(0, 57071), (1, 45785), (2, 18039), (3, 4782), (4, 972), (5, 151), (6, 15)] ... 370 0.7983 126602 [(0, 56774), (1, 45705), (2, 18328), (3, 4656), (4, 976), (5, 145), (6, 16)] ... 380 0.7908 126477 [(0, 57212), (1, 45450), (2, 18109), (3, 4656), (4, 891), (5, 141), (6, 16)] ... 390 0.7919 126308 [(0, 56878), (1, 45781), (2, 17949), (3, 4626), (4, 925), (5, 131), (6, 18)] ... 400 0.7982 126213 [(0, 56666), (1, 45372), (2, 18360), (3, 4786), (4, 874), (5, 134), (6, 20)] ... 410 0.7897 125780 [(0, 56758), (1, 45529), (2, 17854), (3, 4629), (4, 864), (5, 133), (6, 11)] ... 420 0.7894 126118 [(0, 56971), (1, 45709), (2, 17690), (3, 4705), (4, 899), (5, 117), (6, 21)] ... 430 0.7883 125633 [(0, 56779), (1, 45494), (2, 17690), (3, 4678), (4, 852), (5, 125), (6, 14)] ... 440 0.7895 125177 [(0, 56556), (1, 45221), (2, 17799), (3, 4569), (4, 872), (5, 146), (6, 11)] ... 450 0.7862 125515 [(0, 56970), (1, 45118), (2, 17863), (3, 4592), (4, 825), (5, 130), (6, 14)] ... 460 0.7853 125435 [(0, 56765), (1, 45453), (2, 17682), (3, 4606), (4, 797), (5, 115), (6, 16)] ... 470 0.7832 125215 [(0, 56876), (1, 45182), (2, 17691), (3, 4516), (4, 813), (5, 121), (6, 14)] ... 480 0.7858 125176 [(0, 56980), (1, 44921), (2, 17573), (3, 4670), (4, 895), (5, 120), (6, 14)] ... 490 0.7789 124903 [(0, 57166), (1, 44718), (2, 17613), (3, 4445), (4, 820), (5, 125), (6, 14)] ... 500 0.7783 124497 [(0, 56745), (1, 45059), (2, 17346), (3, 4398), (4, 814), (5, 117), (6, 16)] ... 510 0.7813 124533 [(0, 56663), (1, 44985), (2, 17478), (3, 4439), (4, 820), (5, 132), (6, 15)] ... 520 0.7807 124410 [(0, 56703), (1, 44917), (2, 17374), (3, 4392), (4, 863), (5, 138), (6, 21)] ... 530 0.7802 124420 [(0, 56729), (1, 44855), (2, 17426), (3, 4434), (4, 840), (5, 115), (6, 19)] ... 540 0.7814 124181 [(0, 56506), (1, 44864), (2, 17411), (3, 4414), (4, 837), (5, 133), (6, 16)] ... 550 0.7745 124249 [(0, 57085), (1, 44490), (2, 17341), (3, 4423), (4, 781), (5, 109), (6, 18)] ... 560 0.7768 124309 [(0, 56931), (1, 44634), (2, 17434), (3, 4343), (4, 823), (5, 124), (6, 18)] ... 570 0.7753 123791 [(0, 56786), (1, 44586), (2, 17034), (3, 4376), (4, 871), (5, 126), (6, 10)] ... 580 0.7722 123776 [(0, 56845), (1, 44572), (2, 17152), (3, 4283), (4, 778), (5, 129), (6, 16)] ... 590 0.7723 123710 [(0, 56843), (1, 44473), (2, 17187), (3, 4285), (4, 785), (5, 124), (6, 12)] ... 600 0.7724 123686 [(0, 56951), (1, 44265), (2, 17211), (3, 4338), (4, 790), (5, 115), (6, 15)] ... 610 0.7674 123510 [(0, 57082), (1, 44374), (2, 16802), (3, 4348), (4, 778), (5, 111), (6, 13)] ... 620 0.7684 123556 [(0, 57159), (1, 44325), (2, 16799), (3, 4243), (4, 883), (5, 125), (6, 19)] ... 630 0.7698 123254 [(0, 56848), (1, 44321), (2, 16835), (3, 4295), (4, 800), (5, 134), (6, 17)] ... 640 0.7689 123138 [(0, 56787), (1, 44260), (2, 16924), (3, 4255), (4, 767), (5, 130), (6, 13)] ... 650 0.7687 123431 [(0, 57061), (1, 44195), (2, 16936), (3, 4307), (4, 790), (5, 125), (6, 15)] ... 660 0.7672 123241 [(0, 56817), (1, 44443), (2, 16915), (3, 4151), (4, 774), (5, 121), (6, 18)] ... 670 0.7645 122770 [(0, 56828), (1, 44145), (2, 16719), (3, 4183), (4, 770), (5, 109), (6, 14)] ... 680 0.7652 122745 [(0, 56792), (1, 44192), (2, 16616), (3, 4234), (4, 774), (5, 121), (6, 13)] ... 690 0.7663 122849 [(0, 56717), (1, 44196), (2, 16897), (3, 4153), (4, 763), (5, 104), (6, 19)] ... 700 0.7665 122746 [(0, 56780), (1, 44090), (2, 16743), (3, 4191), (4, 796), (5, 128), (6, 18)] ... 710 0.7653 122500 [(0, 56894), (1, 43803), (2, 16586), (3, 4263), (4, 804), (5, 135), (6, 12)] ... 720 0.7602 122483 [(0, 57198), (1, 43570), (2, 16659), (3, 4142), (4, 790), (5, 114), (6, 7)] ... 730 0.7670 122470 [(0, 56826), (1, 43651), (2, 16799), (3, 4247), (4, 808), (5, 123), (6, 14)] ... 740 0.7557 122407 [(0, 57257), (1, 43754), (2, 16462), (3, 4045), (4, 773), (5, 95), (6, 20)] ... 750 0.7617 122109 [(0, 56627), (1, 43891), (2, 16676), (3, 4028), (4, 768), (5, 104), (6, 15)] ... 760 0.7631 122129 [(0, 56715), (1, 43772), (2, 16573), (3, 4164), (4, 767), (5, 116), (6, 21)] ... 770 0.7594 122356 [(0, 57036), (1, 43830), (2, 16431), (3, 4171), (4, 747), (5, 125), (6, 14)] ... 780 0.7626 122112 [(0, 56677), (1, 43878), (2, 16473), (3, 4186), (4, 768), (5, 115), (6, 12)] ... 790 0.7639 121865 [(0, 56503), (1, 43831), (2, 16440), (3, 4142), (4, 810), (5, 116), (6, 20)] ... 800 0.7574 121842 [(0, 56844), (1, 43617), (2, 16465), (3, 4065), (4, 730), (5, 104), (6, 16)] ... 810 0.7544 121915 [(0, 57025), (1, 43730), (2, 16279), (3, 3994), (4, 750), (5, 122), (6, 15)] ... 820 0.7602 121706 [(0, 56702), (1, 43528), (2, 16492), (3, 4102), (4, 736), (5, 126), (6, 17)] ... 830 0.7579 121522 [(0, 56842), (1, 43177), (2, 16628), (3, 3976), (4, 768), (5, 114), (6, 15)] ... 840 0.7583 121800 [(0, 56955), (1, 43291), (2, 16599), (3, 4099), (4, 732), (5, 104), (6, 19)] ... 850 0.7524 121636 [(0, 57143), (1, 43260), (2, 16419), (3, 3969), (4, 725), (5, 108), (6, 11)] ... 860 0.7547 121453 [(0, 56864), (1, 43363), (2, 16322), (3, 4084), (4, 711), (5, 97), (6, 12)] ... 870 0.7534 121207 [(0, 56882), (1, 43189), (2, 16268), (3, 4023), (4, 712), (5, 121), (6, 11)] ... 880 0.7579 121343 [(0, 56512), (1, 43660), (2, 16227), (3, 4059), (4, 763), (5, 109), (6, 11)] ... 890 0.7470 121146 [(0, 57306), (1, 42899), (2, 16168), (3, 3961), (4, 699), (5, 101), (6, 10)] ... 900 0.7500 121301 [(0, 57154), (1, 43148), (2, 16136), (3, 4031), (4, 710), (5, 111), (6, 10)] ... 910 0.7520 121151 [(0, 56669), (1, 43662), (2, 16005), (3, 3964), (4, 724), (5, 115), (6, 11)] ... 920 0.7510 121001 [(0, 56853), (1, 43159), (2, 16196), (3, 3979), (4, 702), (5, 98), (6, 13)] ... 930 0.7509 120862 [(0, 56613), (1, 43382), (2, 16145), (3, 3933), (4, 677), (5, 95), (6, 16)] ... 940 0.7483 120971 [(0, 57154), (1, 42978), (2, 15947), (3, 4050), (4, 720), (5, 110), (6, 12)] ... 950 0.7488 120743 [(0, 56753), (1, 43287), (2, 15940), (3, 3924), (4, 735), (5, 90), (6, 14)] ... 960 0.7510 120863 [(0, 57044), (1, 42704), (2, 16285), (3, 3973), (4, 728), (5, 115), (6, 14)] ... 970 0.7463 120685 [(0, 56773), (1, 43370), (2, 15845), (3, 3872), (4, 735), (5, 84), (6, 6)] ... 980 0.7472 120957 [(0, 57088), (1, 43031), (2, 16103), (3, 3938), (4, 673), (5, 106), (6, 16)] ... 990 0.7467 120735 [(0, 56964), (1, 42985), (2, 16138), (3, 3835), (4, 700), (5, 97), (6, 14)] ... Last fiddled with by GP2 on 2018-12-17 at 06:51 |
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2018-12-17, 07:15
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#61 | ||
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Sep 2003
1010000110012 Posts |
Quote:
The large skew is just in the Mersenne primes themselves. The data tables of factor frequencies just reflect the much smaller "a=2 vs. a=6" heuristic of Wagstaff, and nothing is gained by splitting further according to mod 8 rather than mod 4. If there's some magic going on, it isn't at the small bit lengths where we are capable of finding factors. Maybe the million-digit factors are doing something funny, who knows. Quote:
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2018-12-17, 08:03
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#62 |
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Sep 2003
5·11·47 Posts |
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. Code:
mod 4
149 p=1
176 p=3
mod 8
75 p=1
86 p=3
74 p=5
90 p=7
If we consider only exponents larger than 1200, the numbers are: Code:
mod 4
60 p=1
83 p=3
mod 8
34 p=1
38 p=3
26 p=5
45 p=7
Code:
mod 4
28 p=1
45 p=3
mod 8
17 p=1
21 p=3
11 p=5
24 p=7
Code:
mod 4
11 p=1
24 p=3
mod 8
6 p=1
8 p=3
5 p=5
16 p=7
Code:
mod 4
3 p=1
9 p=3
mod 8
2 p=1
3 p=3
1 p=5
6 p=7
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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2018-12-17, 08:43
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#63 |
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Sep 2003
5·11·47 Posts |
A conjecture!
So, a conjecture:
For Mersenne numbers with an average number of factors, the distribution of p=1 (mod 4) and p=3 (mod 4) is according to Wagstaff's heuristic. Namely, the ones with p=3 have a fractionally larger number of factors on average (I gave estimates at https://www.mersenneforum.org/showthread.php?p=495604 and also provided some data tables in the past few posts). But I conjecture that among the Mersenne numbers with a much smaller than average number of factors, there is a strong clustering into two separate groups. The ones with zero factors, i.e. Mersenne primes, are much more likely to have p=1 (mod 4) and p=1 (mod 8), whereas the ones with very-few-but-non-zero factors are much more likely to have p=3 (mod 4) and perhaps p=7 (mod 8). The first part of the above is old news, we were already talking about the prevalence of p=1 (mod 4) vs. p=3 (mod 4) on this board back in 2003, and others surely noticed it earlier. But I'm not sure if the modulo-8 angle was talked about, or the apparent opposite prevalence of p=3 (mod 4) for fully-factored and probably-fully-factored Mersenne numbers. (Technically, a Mersenne prime is literally fully factored, but we exclude them when we use the term). And I don't think the Wagstaff-prime angle was talked about either, see below. The second part of the above is really only testable when "very few factors" means "small factors plus an enormous cofactor". By contrast, if a Mersenne number is a semiprime with both prime factors comparable in size to each other and to the square root of the Mersenne number, then we'll probably never know it unless it has a tiny exponent in the Cunningham range. There may be a similar clustering effect with Wagstaff numbers that have a much smaller than average number of factors. The ones with zero factors, i.e. Wagstaff primes, are much more likely to have p=3 (mod 4) and p=7 (mod 8). Unfortunately, factoring and PRP cofactor testing of Wagstaff numbers is much shallower so far than for Mersenne numbers, only up to about 56 bits for p < 1M and 58 bits for 1M < p < 2M, and no PRP cofactor testing beyond p=2M. But in any case we currently see very little skew. For exponents of fully-factored Wagstaff numbers with exponents greater than 1000, we have 57 with p=1 (mod 4) and 48 with p=3 (mod 4). Perhaps we will see a meaningful skew develop if deeper factoring and PRP testing is done. Finally, I don't think we can say anything about Mersenne numbers with a much larger than average number of factors, because we don't have any way to identify them. We are only capable of detecting very small factors. We have a table of exponents with the most known factors, but nearly all the exponents in that list have composite cofactors that are 99.9...% as large as the Mersenne number itself, and we can't predict how many or how few prime factors go into that composite cofactor. If the occurrence of factors is a Poisson process, there is no reason to think that the number of prime factors in that enormous cofactor is in any way correlated to the number of factors already known. Last fiddled with by GP2 on 2018-12-17 at 09:42 |
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2018-12-17, 09:19
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#64 | |
"Jeppe"
Jan 2016
Denmark
23·3·7 Posts |
Quote:
The asymptotic behavior which is conjectured to prevail in the long run, does not work by compensating our "lucky" period recently with an unlucky period later on! Instead it works by "drowning" our luck with equal amounts of luckier-than-usual periods and unluckier-than-usual periods until the point where our short period (say, the Mersenne #39 through #51) is negligible. /JeppeSN |
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2018-12-17, 09:31
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#65 | |
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Undefined
"The unspeakable one"
Jun 2006
My evil lair
24×389 Posts |
Quote:
Every one knows that if 6 has not come up recently on the roulette table then it is "due". Bet everything on number 6, you're guaranteed to end up |
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2018-12-17, 14:25
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#66 | ||
Feb 2017
Nowhere
13×359 Posts |
Quote:
Quote:
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