# primorial

The primorial of $n$, or $n\mathrm{\#}$, is the product^{} of the first $n$ consecutive primes, thus:

$$\prod _{i=1}^{n}{p}_{i}$$ |

(${p}_{i}$ is the $i$th prime number^{}).

The first few primorials are 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130; these are listed in A002110 of Sloane’s OEIS. Sometimes the notation $n\mathrm{\#}$ is used to refer to the product of all primes $$, where $\pi $ is the prime counting function (so then $4\mathrm{\#}=6$ rather than 210).

Primorials are used in the classic proof that there are infinitely many primes: assuming that there are exactly $n$ primes and no more, $n\mathrm{\#}+1$ is a number that is not divisible by any of the existing primes, but if that is a prime then it contradicts the initial assumption^{}.

If, in reckoning the sieve of Eratosthenes^{}, one strikes out again numbers that have already been struck off, the sequence^{} of the smallest number struck off $n$ times is precisely the sequence of the primorials.

Any highly composite number (with the exception of 1) can be expressed as a product of primorials in at least one way.

Title | primorial |
---|---|

Canonical name | Primorial |

Date of creation | 2013-03-22 16:00:23 |

Last modified on | 2013-03-22 16:00:23 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A41 |