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

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

$p = 3$

The smallest positive integer $n$ greater than $1$ such that:

$n^{3 - 1} \equiv 1 \pmod {3^2}$

is $8$.

$p = 5$

The smallest positive integer $n$ greater than $1$ such that:

$n^{5 - 1} \equiv 1 \pmod {5^2}$

is $7$.