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 (mathematics), 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 ...

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 prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, 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 stu ...

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

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, ... 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, ... where R_''n'' :=

\tfrac

is a repunit, a number consisting only of ''n'' ones (in

base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of t ...

). There are no other circular primes up to 10²⁵. The only other known examples are

repunit prime 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 ''Recr ...

s, which are circular primes by definition. :R₂ (11), R₁₉, R₂₃, R₃₁₇, R₁₀₃₁, R₄₉₀₈₁, R₈₆₄₅₃, R₁₀₉₂₉₇, R₂₇₀₃₄₃, R_5794777, R_8177207, ... 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 The duodecimal system, also known as base twelve or dozenal, is a positional notation, positional numeral system using 12 (number), twelve as its radix, base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 1, units; ...

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 1 ...

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

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

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)

Absolute Primes
(including circular primes), Numberphile video {{Prime number classes Base-dependent integer sequences Classes of prime numbers