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This is more of my own observations than a question.

Let $P(x)=ax^2+bx+c$. For convenience, assume $a,b,c$ are integral. The question is as follows. Suppose, $P(x)$ is a perfect square (for every integer, or for infinitely many integers, as $x$ sweeps through $\mathbb{R}$ or $\mathbb{Z}$). What can we say about $P(\cdot)$? Does it have to be a square of a linear polynomial?

Case 1: It is clear that, if $a>0$, then $P(x)$ is a perfect square (of an integer), for infinitely many $x\in\mathbb{R}$. Simply solve the associated quadratic.

Case 2: Suppose, $P(x)$ is a perfect square, for infinitely many $x\in\mathbb{Z}$. Is it necessarily true that, $P(x)$ itself must be a square of a degree-1 polynomial, that is, is it true that, $P(x)=(dx+e)^2$, for some $d,e\in\mathbb{R}$? The answer is no. Consider, $P(x)=2x^2+1$. From Pell's equations, it holds that, $2x^2+1=y^2$ infinitely often, and the pairs, $(x_n,y_n)$ can be generated via, $(2+\sqrt{3})^n = x_n+y_n\sqrt{3}$.

Case 3: Suppose, $P(x)$ is a perfect square, for every $x\in\mathbb{Z}$. Is it true that, $P(x)=(dx+e)^2$, for some monic polynomial? Now, it turns out that, the answer is yes. There is an elementary proof of this fact (albeit well-known).

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    $\begingroup$ Special case of general results of Davenport, Lewis & Schinzel that I cite here.. Chase links to that classic paper to find later work. $\endgroup$ – Bill Dubuque Apr 6 at 15:21
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    $\begingroup$ And... What is it you want from us? Do you have a question? Why did you post this to a dedicated question-and-answer site otherwise? $\endgroup$ – Arthur Apr 6 at 15:21
  • $\begingroup$ @BillDubuque: Thanks for the link to DLS result. I guessed such a result must exist, but did not know an explicit source. $\endgroup$ – TBTD Apr 6 at 15:31
  • $\begingroup$ @Arthur : (and those, who upvoted that meaningless comment): First, of all, I already said in the beginning that this is my own observation. If you're unhappy, simply downvote, don't spam. This website is not solely for asking questions, and posting answers (and there have been many instances of it). I wanted to share my own view. Two, if you read carefully (as Bill have done), you'd see that, the last part actually has a potential to be generalized. Anyhow. $\endgroup$ – TBTD Apr 6 at 15:36
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    $\begingroup$ "simply downvote, don't spam" I would actually be delighted if everyone who downvoted on this site could leave a comment describing what they are after. That would not in any way be spam. As for my own comment, I was genuinely curious as to what reason you had to post this, and what you were after, because the clear lack of a question makes your post atypical (not necessarily unique, but atypical). You seem to take offense to this. I do not know why. $\endgroup$ – Arthur Apr 6 at 15:56

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