# Anton’s congruence

For every $n\in \mathbb{N}$ ${\left(n\underset{\xaf}{!}\right)}_{p}$ stands for the product^{} of numbers
between $1$ and $n$ which are not divisible by a given prime $p$. And we set
${\left(0\underset{\xaf}{!}\right)}_{p}=1$.

The corollary below generalizes a result first found by Anton, Stickelberger, and Hensel:

Let ${N}_{0}$ be the least non-negative residue of $n\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{s})$ where $p$ is a
prime number^{} and $n\in \mathbb{N}$. Then

$${\left(n\underset{\xaf}{!}\right)}_{p}\equiv {\left(\pm 1\right)}^{\lfloor n/{p}^{s}\rfloor}\cdot {\left({N}_{0}\underset{\xaf}{!}\right)}_{p}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{s}).$$ |

###### Proof.

We write each $r$ in the product below as $i{p}^{s}+j$ to get

${\left(n\underset{\xaf}{!}\right)}_{p}$ | $=$ | $\prod _{\begin{array}{c}1\le r\le n\\ {p}^{s}\xf7\u0338r\end{array}}}r$ | ||

$=$ | $$ | |||

$\equiv $ | $$ | |||

$\equiv $ | ${\left({p}^{s}\underset{\xaf}{!}\right)}_{p}^{\lfloor n/{p}^{s}\rfloor}\cdot {\left({N}_{0}\underset{\xaf}{!}\right)}_{p}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{s}).$ |

From Wilson’s theorem for prime powers it follows that

$${\left(n\underset{\xaf}{!}\right)}_{p}\equiv \{\begin{array}{cc}{\left({N}_{0}\underset{\xaf}{!}\right)}_{p}\text{if}\hfill & p=2,s\ge 3\hfill \\ {(-1)}^{\lfloor n/{p}^{s}\rfloor}\cdot {\left({N}_{0}\underset{\xaf}{!}\right)}_{p}\hfill & \text{otherwise.}\hfill \end{array}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{s}).$$ |

∎

Title | Anton’s congruence^{} |
---|---|

Canonical name | AntonsCongruence |

Date of creation | 2013-03-22 13:22:49 |

Last modified on | 2013-03-22 13:22:49 |

Owner | Thomas Heye (1234) |

Last modified by | Thomas Heye (1234) |

Numerical id | 10 |

Author | Thomas Heye (1234) |

Entry type | Theorem |

Classification | msc 11A07 |

Related topic | Factorial^{} |