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.
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asked Jun 11, 2016 at 21:04
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2$\begingroup$ That won't be enough, $x^2+2$ (for example) also is never a square. $\endgroup$– Christian RemlingJun 11, 2016 at 21:14
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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 $O(\log N)$. $\endgroup$– KConradJun 11, 2016 at 21:17
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$\begingroup$ @KConrad Are you aware of any example where you have $\Omega(\log N)?$ $\endgroup$– Igor RivinJun 11, 2016 at 22:23
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$\begingroup$ @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 SilvermanJun 11, 2016 at 22:31
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$\begingroup$ @JoeSilverman Yes, in Keith's paper he assumes the polynomial is irreducible... $\endgroup$– Igor RivinJun 11, 2016 at 23:09
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A non-trivial 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.
answered Jun 11, 2016 at 22:34
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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.) $\endgroup$ Jun 12, 2016 at 23:39