PlanetMath: Anton's congruence

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Anton's congruence

(Theorem)

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

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

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

$\displaystyle \left(n\underline{!}\right)_p \equiv \left(\pm 1\right)^{\left\lfloor n/p^s \right\rfloor}\cdot \left(N_0\underline{!}\right)_p \pmod{p^s}.$

Proof. We write each

in the product below as

to get

$\displaystyle \left(n\underline{!}\right)_p$	$\displaystyle =$	$\displaystyle \prod\limits_{\substack{1 \le r \le n\\ p^s \not\div r}} r$
	$\displaystyle =$	$\displaystyle \left( \prod\limits_{\substack{0 \le i \le \left\lfloor n/p^s\rig... ...\lfloor n/p^s\right\rfloor \\ 1\le j \le N_0 \\ p^s \not\div j}} ip^s +j\right)$
	$\displaystyle \equiv$	$\displaystyle \prod\limits_{i=0}^{\left\lfloor n/p^s \right\rfloor -1} \prod\li... ...s \not\div j }} j\cdot \prod\limits_{\substack{j=1 \\ p^s \not\div j}}^{N_0} j)$
	$\displaystyle \equiv$	$\displaystyle \left(p^s\underline{!}\right)_p^{\left\lfloor n/p^s\right\rfloor}\cdot \left(N_0\underline{!}\right)_p \pmod{p^s}.$

From Wilson's theorem for prime powers it follows that

$\begin{displaymath} \left(n\underline{!}\right)_p \equiv \begin{cases} \left(N_0... ...erline{!}\right)_p & \text{otherwise.} \end{cases} \pmod{p^s}. \end{displaymath}$

$\qedsymbol$

"Anton's congruence" is owned by Thomas Heye.

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