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April 17th, 2013, 04:27 PM   #11
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Re: Splitting prime numbers in 2 sets

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by johnr I take it that 2 is a multiprime since it divides 0!+1?
Yes. Its status does not affect any of the questions we're working on, though.
Yeah, it was just an informational query from the peanut gallery!

One more to the point of the discussion: How did you come upon your heuristic?

 April 17th, 2013, 04:28 PM #12 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets I forget the counting function. We can build a counting function for each set. Lot of work. Which one of the sets contains more twin primes (I mean both) ?
April 17th, 2013, 04:40 PM   #13
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha I forget the counting function. We can build a counting function for each set. Lot of work. Which one of the sets contains more twin primes (I mean both) ?
Presumably the multiprimes, which should contain both p and p+2 about 3 times as often as the uniprimes. The uniprime fraction should be roughly
$\frac{1}{2-2e+e^2}\approx25.3\%$
of the total.

April 17th, 2013, 04:43 PM   #14
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by johnr One more to the point of the discussion: How did you come upon your heuristic?
Problems with residues mod p come up fairly often, and the simplest thing you can do is assume each happens with the same probability. This leads to the Hardy-Littlewood conjectures and other things. I wrote up the results so I'd know what to expect then I compared the prediction to the numbers and found excellent agreement.

 April 17th, 2013, 05:36 PM #15 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Splitting prime numbers in 2 sets And here come the big question : is the set of uniprime infinite? I conjecture that the uniprime set is finite Good night!
April 17th, 2013, 06:44 PM   #16
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha And here come the big question : is the set of uniprime infinite? I conjecture that the uniprime set is finite Good night!
For what it's worth (and it ain't much!), I'm betting against your conjecture.

 April 17th, 2013, 06:59 PM #17 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Re: Splitting prime numbers in 2 sets Is there a list of the k < p-1 for at least some of the lower p?
April 17th, 2013, 08:52 PM   #18
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Re: Splitting prime numbers in 2 sets

Quote:
 Originally Posted by Mouhaha is the set of uniprime infinite? I conjecture that the uniprime set is finite
Quote:
 Originally Posted by johnr For what it's worth (and it ain't much!), I'm betting against your conjecture.
I concur with johnr. It would be highly unusual if this set was finite.

April 17th, 2013, 09:22 PM   #19
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Re: Splitting prime numbers in 2 sets

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

 April 17th, 2013, 11:20 PM #20 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 I did one final count to 50,000 finding 32,213 multiprimes. The heuristic suggests about 31,606. So far it's holding up pretty well.

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