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]
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 Rn := 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, R5794777, R8177207, … (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 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, R4A5, R5777, R879B, R198B1, R23175, and R311407.
where 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.
References
- ^ The Universal Book of Mathematics, Darling, David J., 11 August 2004, p. 70, ISBN 9780471270478, retrieved 25 July 2010
- ^ Prime Numbers—The Most Mysterious Figures in Math, Wells, D., p. 47 (page 28 of the book), retrieved 27 July 2010
- ^ a b c Circular Primes, Patrick De Geest, retrieved 4 July 2025
- ^ Sloane, N. J. A. (ed.). “Sequence A293663”. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ The mathematics of Oz: mental gymnastics from beyond the edge, Pickover, Clifford A., 2 September 2002, p. 330, ISBN 9780521016780, retrieved 9 March 2011
External links
- Circular prime at The Prime Glossary
- Circular prime at World of Numbers
- OEIS sequence A068652 a related sequence (the circular primes are a subsequence of this one)
- Circular, Permutable, Truncatable and Deletable Primes
- Absolute Primes (including circular primes), Numberphile video