2.1: Problem 06: Prove that if \(p\) is a prime and \(a^2 \equiv b^2 \bmod p\), then \(p \mid (a + b)\) or \(p \mid (a - b)\).
Proof
Suppose \(p\) is prime and that \(a^2 \equiv b^2 \bmod p\). Then by definition
$$
\begin{align*}
p &\mid a^2 - b^2 \\
p &\mid (a - b)(a + b)
\end{align*}
$$
But \(p\) is prime so either \(p \mid (a- b)\) or \(p \mid (a+b)\). \(\ \blacksquare\)