Circular prime
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its 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.
Known circular primes
The first few circular primes areThe smallest representatives in each cycle of circular primes are
where Rn := is a repunit, a number consisting only of n ones. There are no other circular primes up to 1025.
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.
Properties
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.Other bases
The complete listing of the smallest representative prime from all known cycles of circular primes in base 12 iswhere Rn 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.