Finding all primes $p$ such that $5p + 1$ is a perfect square.

Question

Finding all primes $p$ such that $5p + 1$ is a perfect square. I'm aware a perfect square is a natural number of the form $n^2$ but have no idea as to how to go about this.

Answer 1

$$ n^2 - 1 = (n+1)(n-1) = 5 p $$

Answer 2

More generally, to find all solutions to $pq+1=\square$ where $p$ and $q$ are prime:

$pq = n^2-1 =(n+1)(n-1)$.

If $n=2$, $pq = 3$ which can not be.

If $n > 2$ then either $n+1 = p, n-1 = q$ or $n-1 = p, n+1 = q$.

In the first case, $n = p-1 = q+1$ so $p = q+2, n = q+1$.

Therefore $p$ and $q$ must be a twin prime pair.