# Wilson's Theorem

In number theory, Wilson's Theorem states that if integer $p > 1$ , then $(p-1)! + 1$ is divisible by $p$ if and only if $p$ is prime. It was stated by John Wilson. The French mathematician Lagrange proved it in 1771.

## Proofs

Suppose first that $p$ is composite. Then $p$ has a factor $d > 1$ that is less than or equal to $p-1$. Then $d$ divides $(p-1)!$, so $d$ does not divide $(p-1)! + 1$. Therefore $p$ does not divide $(p-1)! + 1$.

Two proofs of the converse are provided: an elementary one that rests close to basic principles of modular arithmetic, and an elegant method that relies on more powerful algebraic tools.

## Elementary proof

Suppose $p$ is a prime. Then each of the integers $1, \dotsc, p-1$ has an inverse modulo $p$. (Indeed, if one such integer $a$ does not have an inverse, then for some distinct $b$ and $c$ modulo $p$, $ab \equiv ac \pmod{p}$, so that $a(b-c)$ is a multiple of $p$, when $p$ does not divide $a$ or $b-c$—a contradiction.) This inverse is unique, and each number is the inverse of its inverse. If one integer $a$ is its own inverse, then $$0 \equiv a^2 - 1 \equiv (a-1)(a+1) \pmod{p} ,$$ so that $a \equiv 1$ or $a \equiv p-1$. Thus we can partition the set $\{ 2 ,\dotsc, p-2\}$ into pairs $\{a,b\}$ such that $ab \equiv 1 \pmod{p}$. It follows that $(p-1)$ is the product of these pairs times $1 \cdot (-1)$. Since the product of each pair is conguent to 1 modulo $p$, we have $$(p-1)! \equiv 1\cdot 1 \cdot (-1) \equiv -1 \pmod{p},$$ as desired. $\blacksquare$

### Algebraic proof

Let $p$ be a prime. Consider the field of integers modulo $p$. By Fermat's Little Theorem, every nonzero element of this field is a root of the polynomial $$P(x) = x^{p-1} - 1 .$$ Since this field has only $p-1$ nonzero elements, it follows that $$x^{p-1} - 1 = \prod_{r=1}^{p-1}(x-r) .$$ Now, either $p=2$, in which case $a \equiv -a \pmod 2$ for any integer $a$, or $p-1$ is even. In either case, $(-1)^{p-1} \equiv 1 \pmod{p}$, so that $$x^{p-1} - 1 = \prod_{r=1}^{p-1}(x-r) = \prod_{r=1}^{p-1}(-x + r) .$$ If we set $x$ equal to 0, the theorem follows. $\blacksquare$

## Problems

### Introductory

• Let $a$ be an integer such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{23}=\frac{a}{23!}$. Find the remainder when $a$ is divided by $13$.

• If ${p}$ is a prime greater than 2, define $p=2q+1$. Prove that $(q!)^2 + (-1)^q$ is divisible by ${p}$. Solution.
• Let ${p}$ be a prime number such that dividing ${p}$ by 4 leaves the remainder 1. Show that there is an integer ${n}$ such that $n^2 + 1$ is divisible by ${p}$.