Circular prime

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.^[1][2] For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime.^[3]

Circular prime

The numbers generated by cyclically permuting the digits of 19937. The first digit is removed and readded at the right side of the remaining string of digits. This process is repeated until the starting number is reached again. Since all intermediate numbers generated by this process are prime, 19937 is a circular prime.
Named after	Circle
Publication year	2004
Author of publication	Darling, D. J.
No. of known terms	65
First terms	2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79
Largest known term	⁠108177207−1/9⁠
OEIS index	A068652 Circular primes (numbers that remain prime under cyclic shifts of digits)

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).^[3]

Known circular primes

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 := ${\displaystyle {\tfrac {10^{n}-1}{9}}}$ is a repunit, a number consisting only of n ones (in base 10). There are no other circular primes up to 10²⁵.^[3]

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.^[4]

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.^[5]

Other bases

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

2, 3, 5, 7, Ɛ, R₂, 15, 57, 5Ɛ, R₃, 117, 11Ɛ, 175, 1Ɛ7, 157Ɛ, 555Ɛ, R₅, 115Ɛ77, R₁₇, R₈₁, R₉₁, R₂₂₅, R₂₅₅, R_4ᘔ5, R₅₇₇₇, R_879Ɛ, R_198Ɛ1, 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.