# Thread: Prove Congruence

1. ## Prove Congruence

I have to show that if p is a prime number, then the congruence x² ≡ 1 (mod p) has only the solutions x ≡ 1 and x ≡ -1.

2. Originally Posted by page929
I have to show that if p is a prime number, then the congruence x² ≡ 1 (mod p) has only the solutions x ≡ 1 and x ≡ -1.
Assume $x^2 \equiv 1 \text{ mod }p$. That is, p divides $x^2-1$. This is a difference of two squares, and so split it up and realise that $p$ divides one of the two factors, because $p$ is prime...