Solutions to easy Diophantine $8pq +1 = a^{2}$, p and q primes

Show that $p = 3$ and $p = 5$ are the only primes with a maximal $3$ solutions each to $8pq + 1 = a^2$, where $p$ and $q$ are prime.

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This isn't true. For example, with $p=11$, you can take $q = 5$ or $q=23$.
Still no. For $p=5$ you can take $q=2$, $q=3$ or $q = 11$. –  Cocopuffs Jan 19 '13 at 16:43
@user55514 Ok - it looks better now; no counterexamples for primes at least up to $10000$. –  Cocopuffs Jan 19 '13 at 17:20