A permutable prime is a which remains prime on every rearrangement
(permutation) of the digits. For example, 337 is a
permutable because each of 337, 373 and 733 are prime.
Most likely, in base ten the only permutable primes
are 2, 3, 5, 7, 13, 17, 37, 79, 113, 199, 337, their permutations,
and the repunit primes 11, ....
Richert, who may have first studied these primes called them permutable primes [Richert1951], but later they were also called absolute primes [BD1974, Johnson1977].
Obviously permutable primes may not have the digits 2, 4, 6, 8 or 5. Looking modulo 7 we also see they may not have all four of the digits 1, 3, 7, and 9 simultaneously. In fact, looking harder modulo seven we see:
We can gain further information from the following theorem:
If we remove the restriction that permutable primes have at least two distinct digits, then all one digit primes, as well as all repunit primes, would be trivially permutable.
Related pages (outside of this work)