# prime partition

A prime partition^{} is a partition^{} (http://planetmath.org/IntegerPartition) of a given positive integer $n$ consisting only of prime numbers^{}. For example, a prime partition of 42 is 29 + 5 + 5 + 3.

If we accept partitions of length 1 as valid partitions, then it is obvious that only prime numbers have prime partitions of length 1. Not accepting 1 as a prime number makes the problem of prime partitions more interesting, otherwise there would always be for a given $n$, if nothing else, a prime partition consisting of $n$ 1s. Almost as bad, however, is a partion of $n$ into $\lfloor \frac{n}{2}\rfloor $ 2s and 3s.

Both Goldbach’s conjecture and Levy’s conjecture can be restated in terms of prime partitions thus: for any even integer $n>2$ there is always a prime partition of length 2, and for any odd integer $n>5$ there is always a prime partition of length 3 with at most 2 distinct elements.

Assuming Goldbach’s conjecture is true, the most efficient prime partition of an even integer is of length 2, while Vinogradov’s theorem^{} has proven the most efficient prime partition of a sufficiently large composite odd integer is of length 3.

Title | prime partition |
---|---|

Canonical name | PrimePartition |

Date of creation | 2013-03-22 17:28:02 |

Last modified on | 2013-03-22 17:28:02 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 05A17 |

Classification | msc 11P99 |