A circular prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

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 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 (The single-digit primes and

repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreat ...

s are the only members of their respective cycles) is 2, 3, 5, 7, R₂, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R₁₉, R₂₃, R₃₁₇, R₁₀₃₁, R₄₉₀₈₁, R₈₆₄₅₃, R₁₀₉₂₉₇, R₂₇₀₃₄₃, R_5794777 and R_8177207, where R_''n'' is a

prime with ''n'' digits. There are no other circular primes up to 10²³. A type of prime related to the circular primes are the

permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study the ...

s, which are a subset of the circular primes (every permutable prime is also a circular prime, but not necessarily vice versa).

Other bases

The complete listing of the smallest representative prime from all known cycles of circular primes in

base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

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 prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...

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

References

External links

Circular prime
at The Prime Glossary

at World of Numbers * a related sequence (the circular primes are a subsequence of this one)

{{Prime number classes Base-dependent integer sequences Classes of prime numbers