For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$.

Question

For twin primes $p$ and $q$, prove there is an integer $a$ such that $p|(a^2-q)$ if and only if there is an integer $b$ such that $q|(b^2-p)$.

Algebraic substitution using $p=q+2$ and the definition of divisibility seems to go nowhere, are there other properties of twin primes that may aid in this proof?

Answer 1

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 $$\left(\frac{p}{q}\right)=\left(\frac{q}{p}\right)$$ which is exactly the content of the claim