Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared

Theorem
Let $p$ be a prime number.

Then there exists at least one positive integer $n$ greater than $1$ such that:
 * $n^{p - 1} \equiv 1 \pmod {p^2}$

Proof
Hence $p^2 + 1$ fulfils the conditions for the value of $n$ whose existence was required to be demonstrated.