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 kar_bon 2008-11-28 12:15

High weight k's

[center][size=+2]High weight k's[/size][/center]

You got some results for High weight k's? Here you go!

A high weight k has a Nash weight of around 5000 or higher.

Tools for calculating the Nash weight were done by T.Ritschel and can be found [url=http://www.mersenneforum.org/showthread.php?t=7213]here[/url].

Note:
[b]It's very helpful for me if everyone who submit many small primes can check for twins, too![/b]

(Edit the first line from the LLR-input file from ":M:1:2:258" to ":P:1:2:257" and start LLR again.)

 gd_barnes 2008-11-28 12:17

Here are primes for many high-weight 15 k's in the k=400K-1.4M range that currently show zero primes. The search limit was n=10K with the exception of one k that I will show in a separate post.

[code]
k: n's prime
439725: 1, 2, 3, 5, 6, 8, 10, 12, 14, 16, 18, 19, 20, 24, 34, 35, 36, 42,
49, 52, 54, 72, 161, 165, 204, 207, 220, 241, 264, 275, 308, 402,
427, 444, 528, 532, 549, 577, 585, 684, 729, 781, 862, 1042, 1225,
1320, 1809, 1811, 1955, 2149, 2316, 2653, 3077, 3122, 3699, 4460,
4645, 6996, 7536, 8630, 9427 (61 primes)

458055: 1, 2, 3, 4, 7, 10, 12, 14, 22, 35, 39, 44, 52, 56, 61, 105, 190,
210, 217, 254, 260, 313, 371, 404, 410, 664, 686, 854, 1042, 1079,
1156, 1224, 1397, 1505, 1510, 1725, 1852, 1854, 2112, 2260, 2275,
2594, 2689, 2802, 2852, 3152, 3251, 3996, 4432, 4441, 5651, 6826,
8460, 9412 (54 primes)

499455: 1, 3, 5, 8, 10, 12, 19, 27, 29, 36, 38, 40, 42, 55, 64, 65, 70, 88,
92, 94, 103, 145, 158, 169, 171, 185, 188, 196, 200, 209, 222, 236,
317, 318, 397, 493, 497, 898, 916, 964, 967, 1046, 1060, 1084,
1178, 1254, 1417, 1513, 1579, 1580, 1948, 1971, 2149, 2344, 2386,
3772, 4344, 4650, 4694, 5031, 5730, 6126, 6570, 6769, 6805, 7013,
8184, 9338, 9462 (69 primes)

542685: 1, 4, 6, 10, 11, 14, 16, 26, 27, 28, 31, 33, 34, 39, 42, 46, 52, 58,
74, 79, 80, 108, 131, 162, 163, 184, 242, 258, 272, 324, 352, 538,
720, 900, 970, 1072, 1108, 1151, 1212, 1346, 1351, 1491, 1538,
1801, 1975, 2008, 2330, 3622, 3731, 4015, 4723, 5599, 5850, 7195,
7553, 8771, 8782, 9702, 9972 (59 primes)

607815: 1, 3, 4, 6, 7, 10, 11, 13, 16, 17, 18, 19, 36, 39, 40, 43, 44, 49,
57, 60, 77, 79, 100, 107, 108, 136, 137, 149, 151, 154, 178, 188,
193, 208, 214, 220, 240, 246, 247, 274, 360, 361, 364, 387, 406,
528, 591, 607, 706, 834, 1029, 1054, 1368, 1637, 1950, 2069,
2083, 2218, 2398, 2457, 2929, 3489, 3676, 4219, 5076, 5416, 5577,
6777, 7027, 7443, 7607, 8607, 9980 (73 primes)

663465: 1, 3, 4, 5, 6, 8, 11, 12, 15, 17, 18, 24, 29, 32, 42, 44, 53, 58,
64, 72, 139, 152, 155, 174, 176, 180, 200, 207, 209, 222, 449, 491,
512, 531, 791, 845, 868, 875, 1239, 1431, 1462, 1515, 1828, 2012,
2118, 2419, 3179, 3534, 4717, 5667, 6276, 7212, 7488, 8892, 9053,
9133, 9159 (57 primes)

666735: 5, 9, 11, 14, 16, 20, 21, 23, 28, 29, 30, 41, 43, 49, 90, 110, 138,
153, 160, 165, 168, 170, 179, 188, 231, 243, 253, 313, 350, 384,
408, 519, 620, 783, 841, 888, 947, 1189, 1348, 1603, 1613, 1707,
1763, 1997, 2447, 2468, 2474, 2706, 3197, 4260, 4381, 5608, 5763,
5940, 6087, 8577, 9745, 9944 (58 primes)

861735: 2, 3, 4, 6, 7, 12, 14, 18, 25, 26, 27, 29, 30, 35, 37, 43, 53, 68,
69, 80, 98, 104, 109, 110, 124, 127, 133, 162, 194, 200, 209, 278,
296, 306, 323, 359, 364, 414, 415, 470, 500, 507, 615, 672, 844,
990, 1009, 1088, 1569, 1875, 1960, 2183, 2606, 2959, 3006, 3402,
3525, 4760, 4875, 5507, 6233, 7484, 7970, 8489, 9956 (65 primes)

906015: 2, 3, 4, 5, 6, 10, 28, 32, 40, 42, 48, 49, 53, 59, 72, 87, 100, 106,
117, 130, 132, 136, 144, 156, 173, 181, 182, 184, 214, 215, 285,
357, 378, 411, 423, 505, 701, 723, 804, 825, 855, 872, 949, 1118,
1127, 1149, 1191, 1196, 1271, 1460, 1491, 1920, 2553, 3234, 3529,
4330, 4613, 4658, 4706, 5093, 6188, 7466, 8324, 8374 (64 primes)
[/code]

 gd_barnes 2008-11-28 12:41

Here is the most erroneous post in the history of prime searching:
[URL]http://www.mersenneforum.org/showpost.php?p=5422&postcount=153[/URL].

The post states that the person searched k=890505 to n=132K without finding any primes. The post contains 85 errors because that is how many primes there are!! :missingteeth:

Primes for k=890505 up to n=132K:
[code]
1, 2, 4, 6, 11, 18, 19, 23, 25, 32, 41, 45, 48, 68, 86, 90, 100, 115, 116,
125, 146, 166, 174, 184, 191, 232, 235, 249, 268, 273, 274, 311, 321, 417,
664, 725, 765, 770, 991, 1005, 1022, 1142, 1344, 1353, 1512, 1592, 1673,
1688, 1881, 1955, 2394, 2513, 2680, 3377, 3690, 3743, 3902, 4276, 4453,
4880, 4937, 6142, 6754, 6805, 6901, 8768, 9814, 12533, 12734, 12772,
14405, 18155, 20434, 35891, 38963, 41645, 43042, 43436, 52140, 56169,
56182, 62125, 68888, 69112, 109446
[/code]

Moral of the story: Please be specific in the ranges that you have searched and make note of the context of the wording in your post. If the above k had been previously searched prior to this Sept. 3rd, 2003 post, it would have been obvious where the person had started searching. (Likely started searching at n>110K.) Alas, there was no indication in that entire thread that it had ever been searched previously thereby leading one to assume that there are no primes up to n=132K for a k with a weight of ~5000, as currently indicated on rieselprime.org!

Gary

 kar_bon 2008-11-28 12:50

[QUOTE=gd_barnes;151090]Here is the most erroneous post in the history of prime searching:
[URL]http://www.mersenneforum.org/showpost.php?p=5422&postcount=153[/URL].

The post states that the person searched k=890505 to n=132K without finding any primes. The post contains 85 errors because that is how many primes there are!! :missingteeth:
[/QUOTE]

perhaps he searched the wrong base or only n=130k to 132k :grin:

thanks Gary

 gd_barnes 2008-11-28 21:22

[quote=kar_bon;151092]perhaps he searched the wrong base or only n=130k to 132k :grin:

thanks Gary[/quote]

I think he either searched the wrong k or only searched n=110k to 132k. It was surprising how many primes were coming before it just died off for n>70k.

 Thomas11 2008-12-02 13:16

Here are the primes for three high-weight k using Gary's notation. I noticed just now that one of them (k=20887630335) has already been tested by amphoria up to n=88k. Please note the prime for n=1 which is missing in his list.

[CODE]k: n's prime
7924279935: 4, 9, 14, 15, 20, 22, 30, 36, 39, 40, 49, 50, 53, 56, 57, 65, 66,
69, 79, 93, 94, 96, 105, 169, 202, 250, 273, 283, 287, 292, 310,
318, 323, 343, 351, 369, 395, 396, 432, 465, 542, 595, 819, 837,
843, 850, 853, 883, 989, 1009, 1095, 1130, 1138, 1148, 1163, 1249,
1572, 1603, 1866, 1895, 1950, 2423, 3408, 3939, 3955, 4560, 5011,
5093, 5445, 5817, 6802, 7615, 7762, 7848, 8623, 10492, 10980,
12225, 13382, 13774, 15344, 16595, 17078, 17459, 21091, 22288,
22363, 22419, 23371, 24926, 26339, 30957, 32812, 33513, 33997,
39577, 44071, 46648, 46918, 47029, 54264, 57433, 58104, 58940,
62987, 72806, 76401, 82315, 84388, 87252, 87444, 91714, 115885,
117063, 119001, 145419, 164340, 171427, 178596, 181172, 183712,
211725, 213017, 213130, 220548, 223801, 256723 [302k] (127 primes)

20887630335: 1, 4, 6, 7, 8, 9, 25, 28, 30, 31, 32, 44, 45, 52, 55, 58, 59, 62,
68, 71, 78, 86, 96, 130, 134, 140, 144, 148, 181, 183, 188, 230,
247, 263, 311, 316, 351, 352, 354, 412, 825, 865, 881, 964, 988,
992, 1007, 1070, 1175, 1433, 1635, 1783, 1970, 2125, 2151, 2330,
2358, 2648, 2797, 3061, 3462, 4481, 5006, 5178, 6220, 6964, 7270,
7274, 7703, 9283, 9306, 9342, 10147, 10624, 12293, 12624, 13420,
14194, 15365, 16736, 17601, 22337, 27827, 34183, 37218, 37484,
38123, 41407, 41626, 48473, 49084, 51782, 52628, 52711, 53688,
54925, 59548, 69574, 74713, 75237, 79888, 92470, 96735, 100713,
101425, 106891, 113696, 176264 [190k] (108 primes)

37850187375: 2, 5, 13, 16, 19, 20, 41, 56, 57, 64, 66, 74, 90, 97, 100, 107,
130, 137, 145, 147, 149, 171, 172, 178, 208, 241, 258, 301, 302,
318, 330, 331, 386, 439, 460, 462, 526, 529, 556, 558, 563, 625,
626, 628, 666, 699, 797, 804, 890, 907, 928, 968, 1050, 1117,
1247, 1304, 1453, 1584, 1782, 1788, 2115, 2193, 2239, 2371, 2439,
2533, 2789, 2827, 2984, 3044, 3382, 3568, 3633, 3884, 3900, 3909,
4406, 4426, 5833, 6241, 6283, 6325, 6580, 6596, 6714, 7621, 7727,
7968, 8323, 8817, 9002, 9258, 9833, 11742, 11868, 12248, 12294,
12754, 13340, 13719, 15527, 15913, 16323, 16554, 19362, 19802,
20013, 22376, 26187, 26979, 29664, 30135, 34194, 34249, 34615,
38430, 43069, 44140, 45712, 46408, 46526, 51623, 56911, 71097,
72677, 83407, 92569, 95928, 98642, 100905, 102171, 110802,
118475, 118790, 119293, 124258, 127742, 128558, 133092, 146011,
172215, 187837, 211316, 239276, 242178, 248132, 256416, 265996,
273971, 274553, 281322, 314180, 384305, 384898, 410418, 450597,
469548 [470k] (157 primes)
[/CODE]

 Thomas11 2008-12-02 14:09

Here are the primes for another 13 high-weight k, all tested for n=0-10k:

[CODE]k: n's prime
10956100155: 10, 12, 13, 14, 17, 18, 21, 22, 26, 30, 34, 43, 44, 49, 51, 54,
73, 76, 85, 108, 120, 122, 134, 157, 161, 184, 201, 226, 227, 245,
264, 315, 327, 338, 401, 414, 510, 565, 645, 706, 715, 763, 834,
837, 883, 1081, 1099, 1164, 1165, 1192, 1457, 1695, 1710, 1715,
1738, 1912, 2047, 2114, 2142, 2444, 2489, 3125, 3155, 3229, 3446,
3840, 3936, 4314, 4781, 4838, 4950, 5194, 5674, 6700, 7158, 7542,
7798, 7973, 8125, 8629 [10k] (80 primes)

12528535305: 14, 23, 24, 30, 32, 35, 49, 54, 57, 70, 80, 81, 90, 103, 107, 129,
139, 144, 157, 168, 184, 198, 269, 270, 275, 285, 292, 295, 313,
381, 387, 586, 621, 710, 730, 832, 894, 929, 963, 973, 1024, 1112,
1253, 1362, 1365, 1749, 1757, 1852, 1904, 2025, 2217, 3523, 3573,
3798, 4268, 5385, 5967, 6393, 6651, 7436, 7537, 7654, 8835 [10k]
(63 primes)

15065044995: 2, 4, 10, 11, 13, 16, 23, 25, 28, 32, 33, 57, 58, 61, 64, 65, 68,
71, 85, 112, 117, 168, 180, 201, 205, 216, 227, 240, 248, 395,
430, 541, 615, 639, 720, 966, 1031, 1109, 1480, 1528, 1571, 1925,
1995, 2658, 2800, 2821, 2920, 2982, 2983, 3018, 3089, 3145, 3365,
3592, 4193, 4481, 4588, 5165, 6571, 6807, 7547, 8934 [10k]
(62 primes)

16189312425: 1, 2, 3, 12, 13, 14, 15, 17, 19, 25, 34, 57, 95, 97, 109, 120,
136, 177, 178, 217, 279, 288, 375, 396, 413, 433, 506, 537, 577,
593, 711, 717, 750, 777, 784, 909, 997, 1060, 1113, 1185, 1246,
1302, 1499, 1616, 1871, 2783, 3057, 3281, 3336, 3493, 4897, 4940,
5909, 6147, 7784, 8822 [10k] (56 primes)

17809160655: 2, 5, 12, 15, 41, 56, 59, 69, 73, 111, 175, 196, 270, 279, 280,
282, 335, 362, 434, 511, 559, 577, 635, 681, 757, 830, 850, 865,
871, 907, 1024, 1128, 1474, 1523, 1623, 1670, 1912, 1971, 1976,
2014, 2240, 2534, 2998, 3092, 3163, 3293, 4023, 4750, 4935, 5228,
5240, 6197, 6354, 6989, 7067, 7102, 8883, 9476 [10k] (58 primes)

22685496405: 1, 7, 12, 14, 16, 30, 38, 40, 41, 45, 50, 51, 52, 71, 75, 79, 84,
162, 180, 197, 228, 259, 428, 495, 582, 615, 654, 692, 751, 855,
859, 933, 961, 1056, 1188, 1189, 1396, 1621, 1963, 2082, 2187,
2258, 2511, 2645, 2718, 3321, 3984, 4012, 4210, 4265, 4401, 5379,
5787, 5850, 9228 [10k] (55 primes)

25255914255: 1, 2, 5, 8, 11, 17, 19, 34, 42, 44, 46, 47, 48, 51, 53, 63, 80,
85, 88, 90, 102, 106, 107, 114, 120, 132, 183, 192, 194, 209, 222,
227, 263, 265, 291, 300, 362, 392, 402, 492, 497, 518, 547, 583,
636, 652, 807, 933, 991, 1051, 1077, 1163, 1411, 1492, 1594, 1942,
1944, 2081, 2223, 2777, 3256, 3319, 3427, 3491, 3498, 4237, 4824,
4830, 5724, 7272, 7486, 7817, 7862, 8492 [10k] (74 primes)

26860357095: 2, 5, 11, 16, 19, 21, 25, 40, 44, 47, 48, 80, 84, 105, 121, 145,
161, 169, 206, 210, 237, 273, 300, 356, 415, 427, 494, 496, 504,
525, 577, 630, 695, 840, 871, 1029, 1033, 1058, 1340, 1352, 1373,
1375, 1416, 1668, 1699, 1739, 1755, 2111, 2625, 2644, 2744, 2756,
2758, 3544, 3698, 3788, 3834, 4034, 5345, 5719, 6321, 6804, 7387,
9796 [10k] (64 primes)

37609080795: 4, 5, 8, 9, 13, 22, 31, 36, 37, 38, 67, 68, 69, 75, 79, 82, 88,
97, 118, 129, 136, 138, 173, 182, 194, 195, 201, 244, 246, 261,
273, 435, 473, 475, 501, 576, 645, 650, 684, 685, 799, 823, 920,
1140, 1513, 2335, 2385, 2487, 2655, 2788, 2915, 2920, 2923, 3423,
3616, 5253, 5811, 6821, 7258, 7510, 8296, 9136, 9566 [10k]
(63 primes)

38033829405: 1, 5, 9, 10, 13, 14, 15, 17, 23, 25, 29, 41, 49, 53, 54, 55, 75,
86, 94, 100, 113, 116, 121, 133, 140, 151, 163, 170, 236, 267,
293, 315, 374, 532, 579, 611, 612, 613, 631, 688, 693, 757, 775,
782, 858, 884, 891, 946, 1069, 1095, 1189, 1241, 1371, 1479, 1494,
1563, 1878, 2624, 3689, 3761, 3918, 4004, 4187, 4702, 5155, 5648,
5751, 5937, 7861 [10k] (69 primes)

38088930165: 12, 17, 30, 54, 58, 72, 83, 109, 114, 115, 135, 147, 159, 210,
212, 226, 270, 410, 436, 474, 507, 582, 634, 721, 826, 832, 948,
1017, 1117, 1290, 1569, 1594, 1595, 1745, 2019, 2056, 2485, 2540,
2757, 2935, 3110, 3135, 3353, 3553, 3770, 3799, 3876, 3895, 4003,
4094, 4106, 4895, 5633, 5757, 5977, 6022, 6160, 6485, 6855, 7621,
7639, 8121, 8362, 8436, 8742 [10k] (65 primes)

38338029015: 3, 5, 12, 13, 20, 26, 33, 34, 42, 65, 73, 101, 111, 117, 123, 134,
144, 158, 170, 218, 246, 267, 285, 343, 372, 423, 444, 489, 587,
592, 639, 911, 1036, 1096, 1155, 1276, 1281, 1318, 1379, 1585,
1837, 2031, 2514, 2965, 3478, 3512, 3757, 3795, 3883, 4078, 4823,
5370, 5801, 5899, 6092, 6492, 6564, 7046, 7463, 8160, 8676, 8726,
9952 [10k] (63 primes)

38918783475: 4, 29, 30, 37, 45, 67, 71, 96, 125, 147, 150, 176, 177, 236, 249,
260, 323, 336, 353, 358, 366, 377, 488, 572, 590, 601, 634, 836,
1081, 1284, 1298, 1417, 1968, 2180, 2202, 2673, 2784, 3108, 3238,
3245, 3247, 3583, 3676, 3884, 4868, 5161, 5576, 7034, 7035, 7332,
8443, 8692, 9188, 9484 [10k] (54 primes)
[/CODE]

 Thomas11 2008-12-02 14:37

[QUOTE=gd_barnes;151137]I think he either searched the wrong k or only searched n=110k to 132k.[/QUOTE]

Back in 2003 -- when the "old" 15k search was at it's high time -- the entrance level for the Top5000 database was at about n=110000. Since the goal of this project was to find as many primes as possible, most of the users started their search at n=110k or just at the bottom of the Top5000 table. So, I wouldn't call it the "most erroneous post" -- you would probably find many more similar incorrect postings. But I agree with you, that everybody should double check her/his postings for clarity and consistency.

-- Thomas

 kar_bon 2008-12-03 12:00

[QUOTE=Thomas11;151637][CODE]
37850187375:
2, 5, 13, 16, 19, 20, 41, 56, 57, 64, 66, 74, 90, 97, 100, 107, (...)
469548 [470k] (157 primes)
[/CODE][/QUOTE]

[b]so this is the first k ever with more than 150 primes!?![/b]

got anyone other information or is this really the first?

 Thomas11 2008-12-03 13:21

[QUOTE=kar_bon;151772][b]so this is the first k ever with more than 150 primes!?![/b]

got anyone other information or is this really the first?[/QUOTE]

As far as I know, Phil Carmody and Robert Smith were looking such "high density" k-values quite a while ago.
Unfortunately, the information on Phil's page isn't accessible anymore:

[URL="http://fatphil.org/maths/DragRace/proth.html"]http://fatphil.org/maths/DragRace/proth.html[/URL]

 axn 2008-12-03 15:56

Look [URL="http://www.mersenneforum.org/showthread.php?t=9755"]here[/URL], [URL="http://www.mersenneforum.org/showthread.php?t=9632"]here[/URL] and [URL="http://www.mersenneforum.org/showthread.php?t=7945"]here[/URL] for more info. Short answer is no -- there have been other k's with more than 150 primes.

EDIT:- If the question is about _actual_ discovery date, I am not too sure

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