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circular prime




A prime number that remains prime on any cyclic rotation of its digits. An example (in the decimal system) is 1193 because 1931, 9311, and 3119 are also prime. Any one-digit prime is circular by default. In base ten, any circular prime with two or more digits can only contain the digits 1, 3, 7 and 9; otherwise when 0, 2, 4, 5, 6, or 8 is rotated into the units place, the result will be divisible by 2 or 5. The only circular primes known, if we list by just the smallest representative from each cycle, are:

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R19, R23, R317, R1031 and possibly R49081.

These last five are the known repunit primes and probable primes. It is conjectured that there are infinitely many repunit primes, so there should be infinitely many circular primes. But it is highly likely that all circular primes not on the list above are repunits.


Related category

   • PRIME NUMBERS