page_title: Circular prime
hierarchical_section_title: Circular prime
caption_attribution_description: English: The cyclic permutations of the prime number 19937.
context_page_description: 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 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 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, and R₂₇₀₃₄₃, where Rₙ is a repunit 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 primes, which are a subset of the circular primes.
context_section_description: 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. 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 repunits 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, and R₂₇₀₃₄₃, where Rₙ is a repunit 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 primes, which are a subset of the circular primes (every permutable prime is also a circular prime, but not necessarily vice versa).
caption: 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.
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