My Math Forum Splitting prime numbers in 2 sets

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April 18th, 2013, 04:54 AM   #21
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by johnr Is there a list of the k < p-1 for at least some of the lower p?
The first thousand, if I'm not mistaken:
Code:
0, 3, 5, 9, 14, 18, 21, 23, 15, 8, 18, 7, 23, 13, 6, 86, 63, 65, 16, 16, 50, 81, 89, 95, 102, 99, 61, 64, 210, 97, 31, 131, 9, 93, 40, 45, 63, 220, 91, 173, 122, 183, 35, 85, 198, 93, 25, 209, 215, 221, 74, 300, 151, 290, 15, 245, 400, 251, 22, 261, 188, 167, 191, 285, 426, 274, 271, 456, 278, 229, 309, 135, 498, 189, 8, 311, 129, 345, 349, 712, 363, 51, 36, 371, 393, 216, 548, 24, 162, 331, 419, 768, 622, 431, 189, 618, 123, 290, 644, 221, 672, 248, 481, 244, 331, 49, 235, 188, 525, 66, 543, 545, 192, 561, 81, 145, 884, 576, 253, 383, 480, 350, 1204, 1012, 645, 1242, 208, 470, 58, 659, 350, 78, 683, 1258, 328, 645, 725, 72, 1066, 36, 743, 1250, 749, 539, 705, 200, 254, 785, 789, 791, 288, 1232, 1030, 467, 488, 1050, 1290, 539, 375, 627, 833, 26, 536, 861, 617, 741, 891, 1580, 858, 1376, 911, 454, 159, 879, 599, 287, 1086, 1124, 110, 965, 621, 975, 1616, 653, 400, 1001, 832, 802, 1019, 533, 330, 172, 535, 1196, 1055, 1065, 1650, 1071, 293, 1101, 905, 236, 669, 74, 1042, 738, 2080, 221, 809, 1288, 32, 1143, 1155, 576, 456, 1047, 615, 144, 720, 1127, 305, 500, 240, 2132, 1223, 191, 1265, 1032, 1275, 1289, 237, 757, 1258, 72, 124, 1341, 295, 420, 151, 220, 2486, 1067, 2250, 1383, 678, 480, 1401, 1409, 2192, 210, 1534, 1171, 706, 924, 158, 1443, 617, 909, 1207, 1469, 1844, 1481, 402, 1499, 1908, 1505, 448, 132, 1539, 1541, 841, 474, 219, 1583, 2410, 2174, 1595, 2654, 319, 2256, 662, 3138, 1629, 181, 1506, 1115, 342, 1659, 1335, 1402, 1557, 333, 91, 876, 1685, 515, 1695, 127, 2660, 492, 899, 859, 241, 1680, 1755, 2946, 1763, 228, 433, 982, 387, 1773, 938, 349, 527, 1791, 919, 188, 1390, 417, 1005, 844, 1835, 1514, 1845, 2074, 309, 528, 1682, 1925, 1816, 1095, 936, 3834, 1855, 3844, 1035, 2358, 1207, 1983, 50, 2001, 72, 913, 526, 732, 871, 2039, 2045, 311, 761, 1520, 1420, 1378, 68, 981, 3924, 353, 146, 217, 2121, 1215, 1950, 2135, 1256, 2141, 3916, 357, 1790, 2169, 1096, 1704, 932, 2304, 898, 702, 2090, 3124, 502, 2225, 1787, 2954, 1684, 2594, 2253, 3588, 2259, 2261, 2273, 2283, 2291, 945, 2190, 105, 2321, 2325, 87, 2345, 69, 2361, 788, 3222, 897, 2393, 2212, 2416, 2415, 1715, 1022, 2484, 711, 3152, 2329, 2493, 769, 2499, 1478, 2505, 4176, 2511, 1985, 2099, 2260, 127, 924, 86, 4856, 2289, 1677, 657, 458, 158, 2257, 4626, 3704, 4502, 1140, 2675, 625, 2174, 3266, 2709, 1384, 2186, 2721, 638, 3344, 1783, 2741, 1583, 2759, 2721, 2059, 5070, 2811, 1743, 192, 4554, 1897, 2649, 4236, 722, 2855, 3868, 1378, 2871, 1551, 2359, 128, 1391, 2903, 142, 2919, 2921, 2724, 2660, 2933, 2906, 2939, 2912, 4866, 3246, 3260, 625, 675, 426, 293, 1113, 1535, 1868, 6042, 2775, 1970, 193, 3071, 554, 2343, 505, 3862, 1039, 960, 2022, 3131, 3103, 2260, 629, 3155, 58, 57, 439, 3171, 1492, 430, 2492, 4572, 62, 1595, 3213, 1728, 4934, 1262, 1964, 779, 359, 3275, 1000, 3299, 191, 1017, 71, 5398, 6162, 2958, 1289, 3359, 3720, 1266, 3318, 3381, 3395, 824, 3411, 3413, 4916, 3431, 799, 3435, 3441, 1117, 1414, 3473, 4070, 842, 1831, 1571, 3491, 232, 5692, 596, 3539, 765, 2557, 1679, 3575, 486, 3593, 1956, 2337, 3621, 3641, 3653, 137, 3665, 6802, 865, 3569, 3705, 2652, 1915, 938, 1550, 3743, 5076, 3749, 2639, 2822, 582, 774, 771, 2094, 3795, 3803, 3821, 5552, 3632, 1383, 1281, 2162, 3851, 3861, 3170, 3600, 6302, 3879, 3842, 2936, 3095, 1925, 943, 3022, 3939, 2876, 6470, 3953, 3959, 338, 1948, 2464, 1930, 531, 4019, 5514, 280, 3783, 4043, 3307, 4126, 2781, 4061, 2248, 4083, 1522, 127, 4596, 1170, 2347, 1494, 521, 1463, 619, 3802, 317, 4143, 329, 4155, 6752, 6398, 1762, 4193, 604, 1117, 2666, 3054, 2280, 190, 4269, 4271, 4281, 5606, 3316, 5508, 3957, 1080, 3908, 1662, 936, 555, 1408, 1862, 4365, 5878, 3997, 1076, 1763, 4391, 4401, 4403, 4409, 3935, 4415, 4419, 391, 5528, 2619, 1345, 3535, 6428, 6650, 373, 2074, 4505, 1266, 5136, 4533, 4545, 722, 5466, 4563, 1387, 3253, 919, 3514, 4290, 7612, 4613, 3375, 4296, 4533, 2601, 2244, 53, 4655, 4659, 4661, 146, 8254, 8870, 3018, 6400, 4005, 3593, 1171, 3209, 3571, 608, 4731, 4733, 2302, 4739, 4745, 1350, 2998, 1931, 4775, 1778, 8998, 2032, 2036, 3426, 5680, 7690, 421, 2988, 6680, 4869, 4871, 4883, 3449, 4901, 999, 60, 1856, 1877, 4925, 4356, 4929, 4796, 3532, 2970, 4961, 1442, 2030, 462, 5003, 276, 4210, 5019, 1532, 6846, 99, 5049, 160, 3578, 5075, 5079, 1082, 4936, 8550, 5998, 5011, 1254, 5490, 2469, 5135, 1918, 3279, 4643, 3292, 2935, 5594, 1310, 2133, 9694, 5195, 910, 5128, 3236, 5223, 1291, 2987, 5243, 5249, 4854, 608, 5279, 1212, 4795, 1806, 4178, 3573, 5313, 313, 1662, 2903, 5331, 5345, 8346, 3061, 4922, 3990, 5369, 2054, 5385, 3981, 2729, 1948, 880, 1089, 3261, 6078, 2471, 5451, 6248, 4049, 6510, 1034, 5523, 1428, 5529, 3281, 5535, 4118, 2268, 626, 2416, 5565, 604, 9992, 5621, 5625, 7738, 5639, 5643, 5655, 4869, 5675, 458, 11304, 5699, 7160, 6994, 5721, 603, 5733, 2897, 5741, 8612, 257, 5759, 4665, 4785, 5793, 656, 9166, 6210, 8884, 3095, 5849, 1476, 485, 5859, 5865, 9318, 1033, 2380, 3702, 560, 5903, 5913, 1983, 5919, 11706, 712, 4185, 4225, 941, 2294, 5961, 830, 4727, 7042, 10848, 344, 9930, 209, 6005, 10486, 6021, 6035, 1800, 4362, 380, 6071, 120, 1010, 8158, 424, 938, 2353, 4296, 3819, 6131, 2678, 7540, 6161, 1936, 141, 5668, 7566, 30, 3670, 8066, 6225, 10196, 6213, 6251, 4577, 4860, 8396, 2970, 1982, 6305, 6309, 295, 1967, 9974, 6329, 12252, 5224, 11636, 3160, 9570, 4833, 9660, 1990, 6842, 1150, 6399, 8274, 4193, 4653, 8338, 4152, 48, 6449, 194, 6455, 490, 10262, 7906, 10686, 2709, 4384, 6205, 2642, 901, 5523, 766, 3497, 730, 5149, 6563, 6575, 9890, 6585, 6591, 509, 1304, 1236, 3014, 2182, 9810, 190, 2608, 6663, 6665, 3867, 6669, 9002, 3865
Oh cool!

Dang, but you do nice work!

 April 18th, 2013, 05:41 AM #22 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Splitting prime numbers in 2 sets Thanks. Actually I tested the first 50,000 primes but discarded the k-values, so I just re-generated the first thousand for you.
April 18th, 2013, 12:26 PM   #23
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by CRGreathouse Thanks. Actually I tested the first 50,000 primes but discarded the k-values, so I just re-generated the first thousand for you.
More than enough to keep me busy!

 April 18th, 2013, 04:19 PM #24 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Splitting prime numbers in 2 sets There was a program to generate these numbers (not the k-values, but that is easy enough to change) on the OEIS. I didn't use it though, it was too slow. So I wrote a faster version and submitted to the OEIS along with the first 10,000 values. Now maybe the next person who comes across this will be helped.
 April 19th, 2013, 05:53 AM #25 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets Why : 18! mod 23 = 22 and 18! mod 29=28 23 and 29 are multiprimes. The same k make 2 multiprimes. Let us call t(k) the quantity of primes giving p-1 as remains. t(1=2 Is t(k) finite or infinite? Is there a simple way to say that a prime number is multiprime or uniprime (some "belongity" test )?
April 19th, 2013, 09:06 AM   #26
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha Why : 18! mod 23 = 22 and 18! mod 29=28 23 and 29 are multiprimes. The same k make 2 multiprimes. Let us call t(k) the quantity of primes giving p-1 as remains. t(1=2
Why 2? 18! mod 19 = 18, so t(1 >= 3, right?

Quote:
 Originally Posted by Mouhaha Is t(k) finite or infinite?
For any k, t(k) is finite.

April 19th, 2013, 09:17 AM   #27
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Mouhaha Why : 18! mod 23 = 22 and 18! mod 29=28 23 and 29 are multiprimes. The same k make 2 multiprimes. Let us call t(k) the quantity of primes giving p-1 as remains. t(1=2
Why 2? 18! mod 19 = 18, so t(1 >= 3, right?

Quote:
 Originally Posted by Mouhaha Is t(k) finite or infinite?
For any k, t(k) is finite.
Right! t(1=3 and maybe more. It depends on multiprimes to come.
I did not say that.
I was talking about t(k) when n goes to infinite.
That is different.

April 19th, 2013, 09:26 AM   #28
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha I was talking about t(k) when n goes to infinite. That is different.
Are you asking for $\lim_{n\to\infty}t(n)$?

April 19th, 2013, 09:37 AM   #29
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Mouhaha I was talking about t(k) when n goes to infinite. That is different.
Are you asking for $\lim_{n\to\infty}t(n)$?
Yes1 Exactly yes.

 April 19th, 2013, 10:45 AM #30 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Splitting prime numbers in 2 sets The limit does not exist. Probably $\liminf_{n\to\infty}t(n)=1$ and $\limsup_{n\to\infty}t(n)=+\infty.$

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