Definition:Permutable Prime

Definition

A permutable prime is a prime number $p$ which has the property that all anagrams of $p$ are prime.

Sequence

The sequence of permutable primes begins:

$2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R_{19}, R_{23}, R_{317}, R_{1091}, \ldots$

where $R_n$ denotes the repunit of $n$ digits.

The smallest elements of the permutation sets of these are:

$2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, R_{19}, R_{23}, R_{317}, R_{1091}, \ldots$

Also defined as

Some sources insist that a permutable prime must have at least $2$ distinct digits, which excludes the trivial cases of the $1$-digit primes and repunits:

$13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991$

This sequence is A129338 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also known as

A permutable prime can also be referred to as an absolute prime.

The author of this webpage, before learning what its actual name is, toyed with the idea of coining the term permi, inspired by the term emirp.

Also see

• Definition:Repunit Prime

• Results about permutable primes can be found here.