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2020-01-19, 14:58   #7
sweety439

Nov 2016

236010 Posts

Quote:
 Originally Posted by sweety439 For gaps between primes p which 2 is primitive root mod p: A more generalization: gaps of primes p such that znorder(Mod(b,p)) = (p-1)/a, for fixed integers a>=1, b>=2 (for some (a,b) pairs such primes do not exist, e.g. (4,3) and (5,5)).
For the smallest prime p such that znorder(Mod(m,p)) = (p-1)/n, for fixed integers 2<=m<=32, 1<=n<=32 (0 if not exist):

Code:
m\n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
2: 3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593,
3: 2, 11, 67, 13, 41, 61, 883, 313, 271, 431, 5743, 193, 3511, 1583, 2131, 433, 2551, 4177, 8513, 2521, 8779, 683, 10627, 1321, 29851, 1223, 3079, 9661, 49939, 661, 101681, 4129,
4: 0, 3, 0, 17, 0, 31, 0, 73, 0, 151, 0, 433, 0, 631, 0, 337, 0, 127, 0, 241, 0, 331, 0, 601, 0, 4421, 0, 673, 0, 3061, 0, 257,
5: 2, 11, 13, 101, 0, 199, 827, 569, 487, 31, 1453, 181, 7853, 71, 0, 401, 5407, 379, 15277, 761, 1303, 2069, 5107, 409, 0, 1171, 5077, 3109, 1973, 2521, 5023, 449,
6: 11, 19, 7, 5, 31, 139, 463, 97, 37, 101, 353, 241, 4889, 43, 421, 5233, 6563, 1747, 8171, 1901, 11551, 3719, 3037, 409, 28001, 26833, 26407, 11789, 5801, 3931, 48299, 15073,
7: 2, 3, 73, 29, 1031, 19, 43, 113, 883, 311, 353, 1453, 2861, 281, 181, 1873, 409, 1531, 191, 1621, 2311, 419, 14629, 5233, 12251, 7333, 32941, 4397, 11717, 811, 23251, 1409,
8: 3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641,
9: 0, 5, 0, 13, 0, 67, 0, 313, 0, 41, 0, 61, 0, 883, 0, 433, 0, 271, 0, 2161, 0, 683, 0, 193, 0, 1223, 0, 8317, 0, 2131, 0, 769,
10: 7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289,
11: 2, 7, 193, 5, 191, 19, 379, 449, 199, 1301, 2531, 1549, 2081, 547, 61, 1697, 2789, 523, 28843, 661, 1303, 1013, 18539, 2377, 4001, 1847, 31267, 6917, 10499, 1231, 39929, 6689,
12: 5, 23, 19, 37, 271, 13, 29, 193, 487, 11, 89, 373, 521, 421, 211, 5521, 7243, 829, 2129, 1741, 20707, 1453, 10903, 673, 17551, 4993, 12799, 5209, 233, 3181, 25793, 3169,
13: 2, 3, 7, 17, 331, 103, 2017, 673, 1657, 311, 463, 1213, 0, 1303, 271, 337, 1123, 1171, 19001, 61, 421, 7283, 4049, 2617, 1151, 157, 3889, 701, 8237, 601, 71983, 641,
14: 3, 5, 37, 113, 41, 67, 71, 401, 1459, 61, 463, 13, 3121, 659, 1381, 977, 41413, 1009, 1597, 461, 967, 8779, 23369, 12049, 9151, 547, 811, 8233, 132299, 5431, 148367, 2081,
15: 2, 11, 31, 53, 761, 7, 1163, 257, 3691, 311, 991, 1549, 443, 617, 2551, 2417, 1361, 1801, 2129, 3541, 3697, 1123, 12329, 5641, 4651, 2393, 4159, 113, 9629, 1201, 23003, 1249,
16: 0, 3, 0, 5, 0, 31, 0, 17, 0, 151, 0, 109, 0, 631, 0, 113, 0, 127, 0, 1181, 0, 331, 0, 433, 0, 13963, 0, 1709, 0, 3331, 0, 1217,
17: 2, 13, 73, 149, 181, 223, 29, 257, 541, 101, 2003, 229, 1093, 1471, 991, 433, 0, 883, 2851, 1361, 3361, 1409, 19183, 3673, 13901, 3719, 7723, 8093, 6091, 2371, 10789, 1889,
18: 5, 7, 13, 73, 131, 79, 1667, 41, 19, 311, 3917, 1201, 443, 113, 1381, 17, 1259, 199, 229, 2801, 1429, 881, 1427, 1153, 18701, 599, 12853, 6833, 20939, 2671, 19469, 3361,
19: 2, 3, 97, 101, 131, 307, 1303, 233, 271, 1291, 199, 277, 859, 197, 691, 1217, 12037, 487, 24967, 1901, 1009, 8999, 2393, 4561, 4951, 5227, 6373, 8513, 56957, 151, 14447, 2753,
20: 3, 11, 7, 29, 0, 151, 197, 521, 577, 71, 617, 61, 1873, 491, 0, 1489, 307, 19, 7753, 661, 127, 4049, 9293, 1129, 0, 859, 3673, 3221, 44777, 691, 8123, 929,
21: 2, 37, 13, 5, 11, 43, 953, 337, 433, 461, 199, 1129, 599, 211, 661, 881, 3877, 1747, 14897, 3301, 0, 1277, 52901, 1801, 14551, 30707, 2971, 14197, 34337, 1171, 41231, 1697,
22: 5, 3, 43, 13, 241, 7, 631, 521, 73, 461, 23, 613, 157, 127, 5791, 433, 10337, 2647, 37013, 401, 4201, 947, 17021, 97, 12101, 3407, 15013, 6329, 14153, 1381, 12959, 353,
23: 2, 7, 31, 29, 71, 103, 239, 233, 163, 11, 859, 1093, 53, 911, 271, 1153, 7039, 2719, 25423, 461, 211, 1013, 5843, 3889, 1901, 79, 57349, 1933, 13399, 2131, 17299, 4129,
24: 7, 5, 61, 29, 131, 67, 127, 457, 613, 311, 199, 2617, 79, 379, 991, 241, 4999, 307, 12541, 6581, 8527, 23, 11777, 1009, 1451, 4967, 22303, 2381, 349, 1321, 5023, 4801,
25: 0, 3, 0, 29, 0, 13, 0, 569, 0, 31, 0, 181, 0, 71, 0, 401, 0, 379, 0, 641, 0, 1453, 0, 409, 0, 1171, 0, 3109, 0, 2851, 0, 8609,
26: 3, 11, 151, 5, 31, 19, 547, 313, 1657, 1031, 859, 37, 6397, 3823, 181, 337, 4421, 3853, 4409, 7741, 757, 2311, 37307, 8161, 3701, 2393, 19441, 1597, 1567, 5101, 23561, 4001,
27: 2, 11, 7, 0, 41, 37, 1289, 0, 307, 431, 9857, 13, 7853, 1583, 1051, 0, 7481, 73, 8513, 0, 883, 683, 14813, 313, 38501, 1223, 271, 0, 59393, 661, 101681, 0,
28: 5, 3, 61, 53, 601, 199, 127, 449, 1423, 281, 4093, 1117, 3719, 29, 631, 113, 4999, 613, 23447, 541, 547, 6359, 6211, 6073, 14851, 4733, 4159, 6469, 33641, 4561, 1861, 6113,
29: 2, 5, 31, 13, 61, 7, 617, 1289, 541, 571, 727, 181, 2549, 673, 3121, 2609, 1259, 3061, 2927, 11981, 757, 67, 12743, 7321, 11701, 313, 16417, 12853, 0, 1831, 8123, 12577,
30: 11, 7, 73, 17, 991, 19, 1289, 257, 163, 71, 67, 277, 53, 1163, 31, 113, 1259, 613, 7069, 461, 337, 947, 9293, 409, 401, 1171, 3673, 29, 52259, 241, 14323, 10337,
31: 2, 3, 13, 5, 191, 271, 659, 977, 37, 541, 5237, 349, 4759, 911, 4111, 1217, 2143, 2683, 2129, 3221, 2689, 3499, 2531, 2857, 7901, 1613, 11827, 2437, 45821, 571, 40487, 577,
32: 3, 7, 43, 113, 11, 223, 1163, 73, 397, 41, 1013, 1753, 4733, 673, 691, 257, 1429, 127, 6043, 281, 33013, 6337, 18539, 1777, 251, 14327, 5347, 2857, 72269, 31, 683, 2593,
For more values (2<=m<=64, 1<=n<=64), see https://de.wikipedia.org/w/index.php...ldid=195976169

Last fiddled with by sweety439 on 2020-01-19 at 15:13