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Circular primes
Circular primes are primes where all numbers produced by cyclically shifting the digits of those primes in a given base are also prime in that given base. Primes less than the base and repunit primes are trivially circular primes.
For example, the prime 197 is a circular prime in base 10 since 719 and 971 are also primes. 199 is also a circular prime since 919 and 991 are also primes. But 211 is not a circular prime because 121 = 11 2. In base 10, with the exception of 5 itself, no prime with the digit 5 is a circular prime.
In even bases, circular primes don't contain even digits. In binary this means that only repunits (the Mersenne primes) are circular primes.
All absolute primes are circular primes, but not all circular primes are absolute primes. For example, 197 is circular but not absolute since 791 = 7 × 113.
Base | Circular primes | |
2 | 3, 7, 31, 127, 8191 | A000668 |
3 | 2, 5, 13, 1093, 797161, 3754733257489862401973357979128773 | A272106^{[1]} |
4 | 2, 3, 5, 7, 23, 383 | |
5 | 2, 3, 7, 13, 19, 31, 167, 239, 1069, 1249 | |
6 | 2, 3, 5, 7, 11, 43, 71 | |
7 | 2, 3, 5, 11, 13, 17, 19, 41, 79, 89, 97, 131, 439, 479, 509, 571, 619, 677, 853, 977, 1021, 1657, 2801, 3001, 4447, 9587 | |
8 | 2, 3, 5, 7, 13, 29, 31, 47, 73, 607, 719, 751, 1021, 1759 | |
9 | 2, 3, 5, 7, 11, 13, 17, 23, 43, 71, 101, 149, 211, 233, 647, 971, 1123, 1361, 1429, 1697, 3371, 5741, 8681 | |
10 | 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933 | A016114 |
11 | 2, 3, 5, 7, 13, 17, 19, 29, 31, 37, 43, 53, 61, 73, 97, 139, 163, 223, 241, 281, 317, 401, 433, 439, 479, 677, 701, 967, 1621, 1627, 1879, 2011, 2017, 2381, 3373, 3581, 3593, 3613, 3803, 4591, 4639, 4679, 4957, 5279, 5927, 6257, 6269, 7621, 9173 | |
12 | 2, 3, 5, 7, 11, 13, 17, 67, 71, 157, 163, 167, 233, 283, 2543, 9431 | |
13 | 2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 47, 71, 73, 89, 101, 103, 127, 193, 211, 239, 241, 271, 373, 397, 401, 433, 461, 557, 569, 607, 617, 619, 659, 673, 811, 829, 1009, 1283, 1297, 1321, 2459, 2689, 2729, 2767, 2803, 2857, 2909, 2953, 3019, 3041, 3257, 3271, 3359, 3373, 3461, 3571, 3593, 3623, 3691, 3803, 3931, 4133, 4373, 4783, 4973, 5113, 5147, 5171, 5573, 5711, 5783, 5807, 6053, 6211, 6247, 6353, 7193, 7253, 7331, 7433, 7549, 7577, 7727, 7757, 8017, 8111, 8363, 8419, 8753, 8779, 8783, 9521, 9547, 9551, 9613, 9769 | |
14 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 47, 53, 79, 137, 139, 167, 211, 643, 2963, 3407, 3463, 3911, 3919, 5059, 5087 | |
15 | 2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 67, 73, 113, 131, 241, 269, 433, 487, 569, 577, 673, 971, 1013, 1019, 1103, 1693, 1753, 2699, 3671, 3727, 3767, 3803, 3821, 4297, 4339, 4409, 5119, 5347, 6469, 6473, 9341 | |
16 | 2, 3, 5, 7, 11, 13, 17, 23, 31, 53, 59, 61, 89, 191, 277, 283, 311, 317, 373, 509, 857, 887, 1019, 1021, 1373, 1439, 1471, 1979, 3037, 6043, 6547, 7507, 7549, 7603, 8053 | |
17 | 2, 3, 5, 7, 11, 13, 23, 31, 37, 47, 59, 61, 73, 79, 101, 113, 131, 149, 151, 163, 181, 307, 349, 683, 719, 821, 1117, 1129, 1913, 1933, 2203, 2311, 3109, 5309, 5519, 5869, 6131, 6277, 6299, 6311, 6551, 6991, 7069, 7927, 7949, 8011, 8221, 8641, 8713, 9049, 9137, 9319, 9497, 9511, 9613, 9649 | |
18 | 2, 3, 5, 7, 11, 13, 17, 19, 29, 97, 103, 107, 139, 353, 359, 571, 1721, 2477, 2591, 3779, 7481, 8423, 9431, 9601 | |
19 | 2, 3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 47, 61, 67, 71, 73, 83, 89, 103, 107, 109, 113, 149, 151, 167, 181, 223, 227, 263, 283, 383, 401, 523, 601, 619, 661, 719, 773, 907, 997, 1021, 1063, 1213, 1223, 1259, 1283, 1289, 1307, 1367, 1433, 1531, 1549, 1567, 1741, 1949, 2011, 2029, 2039, 2099, 2143, 2389, 2677, 2713, 2843, 3169, 3967, 4271, 4273, 4289, 4651, 7687, 7723, 8089, 8377, 8387, 9013, 9203, 9323, 9697 | |
20 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 71, 73, 79, 151, 157, 191, 193, 197, 199, 233, 277, 359, 421, 1277, 1429, 1549, 2953, 8423, 8429, 8599, 8627, 9587 |
See also
- ↑ This is actually a selection of absolute primes.