Squaring an integer quadratic

Question

Let $Q(p)=37p^2-47p+4$. How does one go about finding all primes $p$ such that $Q(p)$ is a square of an integer?

A bonus question. Is there a regular method to construct integer triples $(a,b,c)$ such that $Q(p)=ap^2+bp+c$ is a square for a finite number of primes?

Yes, $a=1$, $b=2$ and $c=0$. Then $Q(p)=(p+1)^2-1$ is square for only finitely many $p$. — Dietrich Burde, Sep 07 '13 at 18:58

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

1

The Diophantine equation $n^2=37p^2-47p+4$ can be solved, as can be all binary quadratic Diophanite equations, see How to solve inhomogeneous quadratic forms in integers?. In your case, start with $p=3,23,...$.

edited Apr 13 '17 at 12:19

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answered Sep 07 '13 at 19:01

Dietrich Burde

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Squaring an integer quadratic

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