# permutable prime

Given the base $b$ representation of a prime number^{} $p$ as ${d}_{k},\mathrm{\dots},{d}_{1}$ with

$$p=\sum _{i=1}^{k}{d}_{i}{b}^{i-1},$$ |

if each possible permutation^{} of the digits still represents a prime number in that base, then $p$ is said to be a *permutable prime ^{}*. 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 ${\pi}_{P}(n)$ to count how many permutable primes there are below $n$, it is obvious that ${\pi}_{P}(b-1)=\pi (b-1)$, where $\pi (n)$ 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 primes^{}). Richert proved in 1951 that in the range $$ the only base 10 permutable primes are repunit primes; it is conjectured that this is also true above that range.

## References

- 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 |