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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