Let $p$ and $q$ be a pair of twin primes, such that $q = p + 2$.
Prove the following:
$\exists$ an integer $a$ such that $p \mid (a^2 - q)$ $\iff$ $\exists$ an integer $b$ such that $q \mid (b^2-p)$.
I understand I have to show $a^2 \equiv q\ (\operatorname{mod} p)$ $\iff$ $b^2 \equiv p\ (\operatorname{mod}q)$ but I do not know where to go from here.
I was thinking of applying Wilson's Theorem but I'm uncertain.
In a twin-prime pair $\ p\ $ and $\ q\ $, necessarily either $\ p\ $ is of the form $\ 4k+1\ $ and $\ q\ $ of the form $\ 4k+3\ $ or vice versa. In this case , we have $$(\frac{p}{q})=(\frac{q}{p})$$ which is exactly the content of the claim.