A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime.
A type of prime related to the circular primes are the permutable primes, which are a subset of the circular primes (every permutable prime is also a circular prime, but not necessarily vice versa).
The first few circular primes are
The smallest representatives in each cycle of circular primes are
where R<sub>n</sub> := is a repunit, a number consisting only of n ones (in base 10). There are no other circular primes up to 10<sup>25</sup>.
The only other known examples are repunit primes, which are circular primes by definition.
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)
where R<sub>n</sub> is a repunit prime in base 12 with n digits. There are no other circular primes in base 12 up to 12<sup>12</sup>.
In base 2, only Mersenne primes can be circular primes, since any 0 permuted to the one's place results in an even number.