Anton’s congruence

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

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

Let $N_{0}$ be the least non-negative residue of $n\pmod{p^{s}}$ where $p$ is a prime number and $n\in\mathbb{N}$ . Then

$\left(n\underline{!}\right)_{p}\equiv\left(\pm 1\right)^{\left\lfloor n/p^{s}% \right\rfloor}\cdot\left(N_{0}\underline{!}\right)_{p}\pmod{p^{s}}.$

Proof.

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

$\displaystyle\left(n\underline{!}\right)_{p}$	$\displaystyle=$	$\displaystyle\prod\limits_{\begin{subarray}{c}1\leq r\leq n\\ p^{s}\not\div r\end{subarray}}r$
	$\displaystyle=$	$\displaystyle\left(\prod\limits_{\begin{subarray}{c}0\leq i\leq\left\lfloor n/% p^{s}\right\rfloor-1\\ 1\leq j<p^{s}\\ p^{s}\not\div j\end{subarray}}ip^{s}+j\right)\left(\prod\limits_{% \begin{subarray}{c}i=\left\lfloor n/p^{s}\right\rfloor\\ 1\leq j\leq N_{0}\\ p^{s}\not\div j\end{subarray}}ip^{s}+j\right)$
	$\displaystyle\equiv$	$\displaystyle\prod\limits_{i=0}^{\left\lfloor n/p^{s}\right\rfloor-1}\prod% \limits_{\begin{subarray}{c}1\leq j<p^{s}\\ p^{s}\not\div j\end{subarray}}j\cdot\prod\limits_{\begin{subarray}{c}j=1\\ p^{s}\not\div j\end{subarray}}^{N_{0}}j)$
	$\displaystyle\equiv$	$\displaystyle\left(p^{s}\underline{!}\right)_{p}^{\left\lfloor n/p^{s}\right% \rfloor}\cdot\left(N_{0}\underline{!}\right)_{p}\pmod{p^{s}}.$

From Wilson’s theorem for prime powers it follows that

$\left(n\underline{!}\right)_{p}\equiv\begin{cases}\left(N_{0}\underline{!}% \right)_{p}\text{if}&p=2,s\geq 3\\ (-1)^{\left\lfloor n/p^{s}\right\rfloor}\cdot\left(N_{0}\underline{!}\right)_{% p}&\text{otherwise.}\end{cases}\pmod{p^{s}}.$

∎

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