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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 $q=23$.

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Ok Cocopuffs you´re right. I should have asked: Show that 3 is the only prime p such that there are 3 primes q that satisfy 8pq +1 =a ^2. –  user55514 Jan 19 '13 at 16:07
    
Still no. For $p=5$ you can take $q=2$, $q=3$ or $q = 11$. –  Cocopuffs Jan 19 '13 at 16:43
    
You are again right Cocopuffs and killing me sofly with your words (that´s sense of humor and just a song). I should have excluded 5 also. –  user55514 Jan 19 '13 at 17:11
    
@user55514 Ok - it looks better now; no counterexamples for primes at least up to $10000$. –  Cocopuffs Jan 19 '13 at 17:20
    
Yes, Cocopuffs, no counterexample up to 100000, as for me, with Pari gp, and also an elemental short proof that 3 and 5 are the only such primes with as much as 3 solutions. –  user55514 Jan 19 '13 at 17:33
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