Wilson's Theorem/Corollary 1

Theorem

Let $p$ be a prime number.

Then $p$ is the smallest prime number which divides $\paren {p - 1}! + 1$.

Proof

From Wilson's Theorem, $p$ divides $\paren {p - 1}! + 1$.

Let $q$ be a prime number less than $p$.

Then $q$ is a divisor of $\paren {p - 1}!$ and so does not divide $\paren {p - 1}! + 1$.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-3}$ Wilson's Theorem: Exercise $1$