Are there any primes which remain prime on any cyclic rotation of their digits? Yes: 1193, 1931, 9311 and 3119 are all prime. Such primes are called circular primes (or decimal circular primes because this property clearly depends on the radix in which we write the numbers).
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 we rotate a 0, 2, 4, 5, 6, or 8 into the units place, the result will be divisible by 2 or 5.
There are not many known circular primes. In fact, below we list all of the known circular primes by just listing the smallest representative from each cycle, that is, we list just 1193, not 1193, 1931, 9311 and 3119.
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 (a PRP)These last five are the known repunit primes and probable-prime. 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 pages (outside of this work)