Circular prime

The first few circular primes are

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, ... (sequence A068652 in the OEIS)

The smallest representatives in each cycle of circular primes are

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R₁₉, R₂₃, ... (sequence A016114 in the OEIS)

where R_n := is a repunit, a number consisting only of n ones (in base 10). There are no other circular primes up to 10²⁵.

The only other known examples are repunit primes, which are circular primes by definition.

R₂ (11), R₁₉, R₂₃, R₃₁₇, R₁₀₃₁, R₄₉₀₈₁, R₈₆₄₅₃, R₁₀₉₂₉₇, R₂₇₀₃₄₃, R_5794777, R_8177207, ... (sequence A004023 in the OEIS)

It is conjectured that there are only finitely many non-repunit circular primes.

A circular prime with at least two digits can only consist of combinations of the digits 1, 3, 7 or 9, because having 0, 2, 4, 6 or 8 as the last digit makes the number divisible by 2, and having 0 or 5 as the last digit makes it divisible by 5.

The complete listing of the smallest representative prime from all known cycles of circular primes in base 12 is (using A and B for ten and eleven, respectively)

2, 3, 5, 7, B, R₂, 15, 57, 5B, R₃, 117, 11B, 175, 1B7, 157B, 555B, R₅, 115B77, R₁₇, R₈₁, R₉₁, R₂₂₅, R₂₅₅, R_4A5, R₅₇₇₇, R_879B, R_198B1, R₂₃₁₇₅, and R₃₁₁₄₀₇.

where R_n is a repunit prime in base 12 with n digits. There are no other circular primes in base 12 up to 12¹².

In base 2, only Mersenne primes can be circular primes, since any 0 permuted to the one's place results in an even number.