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For Wilson's theorem, if $p$ is prime then $(p-1) \equiv -1 \mod{p}$ and if not $0 \mod{p}$ except for $p=4$, is $p$ and integer or natural number?

Studying Wilson's theorem for Double, Hyper, Sub and Double factorials and I have begun by stating what Wilson's theorem actually is.

However is $p \in \mathbb{Z}$? Or is $p \in \mathbb{N}$?

Andrews
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Keighleyite
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1 Answers1

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I assume you mean $(p-1)!$

It is not a loss of generality to assume $p \in \mathbb{N}$, as primes are unique upto multiplication by units (in this case $\pm 1$). The factorial function is not defined at negative integers. However, you may restate the theorem for $-p$ as:

$(-1)(-2)...(-p+1)$ is congruent to $-1$ modulo $-p$ if $-p$ is prime and $0$ otherwise (except $-p = -4$).

More specifically, $p$ and $-p$ generate the same ideal in $\mathbb{Z}$, so the quotient fields are isomorphic.

vxnture
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