ARSE

Tue 05 Nov 2024

Circular Prime Days

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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, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R_{19}, R_{23}, R_{317} and R_{1031},
where R_{n} is a repunit prime with n digits.

There are no other circular primes up to 10^{23}.