I will prove when 'n' is an odd prime (so we can invoke Wilson's Theorem).
Let p be an odd prime. Then by Wilson's Theorem:
Since we can freely subtract p from each term in the second half of the previous equation. This gives us:
Now note that in the second half of the right hand side there are '-1's' (pardon my notation abuse). Pulling out the '-1's' we have the result.
If then . Hence, by our previous result, . Therefore has a solution.