Sufficient Condition for 5 to divide n^2+1

Theorem
Let:

where $\nmid$ denotes non-divisibility.

Then:
 * $5 \divides n^2 + 1$

where $\divides$ denotes divisibility.

Proof
We have that:

So either:
 * $n \equiv 2 \pmod 5$

or:
 * $n \equiv 3 \pmod 5$

and so:

or: