permutable prime

Given the base b representation of a prime numberMathworldPlanetmath p as dk,,d1 with


if each possible permutationMathworldPlanetmath of the digits still represents a prime number in that base, then p is said to be a permutable primeMathworldPlanetmath. For example, in base 10, the prime 337 is a permutable prime since 373 and 733 are also prime. The known base 10 permutable primes are listed in A003459 of Sloane’s OEIS.

If we define πP(n) to count how many permutable primes there are below n, it is obvious that πP(b-1)=π(b-1), where π(n) is the standard prime counting function.

When 2|b, a search for permutable primes can safely exclude any primes whose base b representation includes digits that are individually even. In a trivial sense, all repunit primes are also permutable primes. This means that in binary, the only permutable primes are repunits (that is, the Mersenne primesMathworldPlanetmath). Richert proved in 1951 that in the range 991<p<10175 the only base 10 permutable primes are repunit primes; it is conjectured that this is also true above that range.


  • 1 H. E. Richert, ”On permutable primtall,” Unsolved Norsk Matematiske Tiddskrift, 33 (1951), 50 - 54.
Title permutable prime
Canonical name PermutablePrime
Date of creation 2013-03-22 16:13:42
Last modified on 2013-03-22 16:13:42
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A63
Synonym absolute prime