# 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

 $\ds p^2$ $\equiv$ $\ds 0$ $\ds \pmod {p^2}$ $\ds 1$ $\equiv$ $\ds 1$ $\ds \pmod {p^2}$ $\ds \leadsto \ \$ $\ds p^2 + 1$ $\equiv$ $\ds 0 + 1$ $\ds \pmod {p^2}$ Modulo Addition is Well-Defined $\ds$ $\equiv$ $\ds 1$ $\ds \pmod {p^2}$ $\ds \leadsto \ \$ $\ds \paren {p^2 + 1}^{p - 1}$ $\equiv$ $\ds 1^{p - 1}$ $\ds \pmod {p^2}$ Congruence of Powers $\ds$ $\equiv$ $\ds 1$ $\ds \pmod {p^2}$

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

$\blacksquare$

## Examples

### $p = 3$

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 $7$.