permutable prime

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:

Theorem
Every permutable prime is a near-repdigit, that is, it is a permutation of the integer

Bn(a,b) = aaa...aab

where a and b are distinct digits from the set {1, 3, 7, 9}.

We can gain further information from the following theorem:

Theorem
Let Bn(a,b) be a permutable prime and let p be a prime such that np. If 10 is a primitive root of p, and p does not divide a, then n is a multiple of p−1.

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)

References:

BD1974
T. Bhargava and P. Doyle, "On the existence of absolute primes," Math. Mag., 47 (1974) 233.  MR 49:10630
Caldwell87a
C. Caldwell, "Permutable primes," J. Recreational Math., 19:2 (1987) 135--138. [Discusses permutable primes such as 733 in base 10, and 742 in base 13.]
Johnson1977
A. Johnson, "Absolute primes," Math. Mag., 50 (1977) 100--103.
Mavlo1995
Mavlo, Dmitry, "Absolute prime numbers," The Mathematical Gazette, 79:485 (1995) 299--304.
Richert1951
H. E. Richert, "On permutable primtall," Norsk Matematiske Tiddskrift, 33 (1951) 50--54.