Consider a quadratic polynomial $p(x) \in \mathbb{Z}[x],$ say $$p(x) = a x^2 + b x + c.$$ The question is: is there an asymptotic estimate for the number of integral $x$ in $[N, N]$ for which $p(x)$ is the square of an integer? Obviously, the polynomial $x^2  1$ is never a square, but perhaps assuming that the leading coefficient is positive tells us that there is at least some reasonable lower bound.

2$\begingroup$ That won't be enough, $x^2+2$ (for example) also is never a square. $\endgroup$ – Christian Remling Jun 11 '16 at 21:14

1$\begingroup$ For an upper bound (not asymptotic estimate), see Lemma 8 of math.uconn.edu/~kconrad/articles/hlconst.pdf. Actually, the case you are asking about (degree $2$ and square values) is at the end of the proof and will refer you to another paper to see an upper (not lower) bound is $O(\log N)$. $\endgroup$ – KConrad Jun 11 '16 at 21:17

$\begingroup$ @KConrad Are you aware of any example where you have $\Omega(\log N)?$ $\endgroup$ – Igor Rivin Jun 11 '16 at 22:23

$\begingroup$ @IgorRivin Presumably when you ask for an example with $\Omega(\log N)$, you want to rule out the case that $p(x)$ is itself a square in $\mathbb Z[x]$. $\endgroup$ – Joe Silverman Jun 11 '16 at 22:31

$\begingroup$ @JoeSilverman Yes, in Keith's paper he assumes the polynomial is irreducible... $\endgroup$ – Igor Rivin Jun 11 '16 at 23:09
A nontrivial answer for your comment/question for an example giving $\Omega(\log N)$, take $p(x)=2x^2+1$. This gives a Pell equation $2x^2+1=y^2$, and taking powers of the fundamental unit will, I believe, give you exactly the $\Omega(\log N)$ behavior that you want. More generally, this should work for lots of polynomials with real quadratic roots.

2$\begingroup$ In fact, as long as the leading coefficient is positive and not square, the theory of "Pell equations" guarantees that once there is one solution there are infinitely many and the number of solutions with $x\leq N$ is asymptotic to a multiple of $\log N$. (If the leading coefficient is a positive square then there are finitely many solutions unless $p$ is the square of a linear polynomial.) $\endgroup$ – Noam D. Elkies Jun 12 '16 at 23:39