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As a result of the discussion in this question Quadratic residues and squares of odd numbers, @Mark Bennet asked me to open this question. I have a quadratic expression $14144x^2+3872x+265$. How can I prove that this is never a perfect square using modular arithmetic or otherwise? In general how can any expression $ax^2+bx+c$ be checked if it is an odd perfect square or not?

Wilbur
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  • Note that setting $y=8x$ simplifies the expression significantly to $p(y)=221y^2+484y+265$. I concur with @auscrypt's answer that the general solution here goes back to the generalised Pell equation. – Mark Bennet May 29 '19 at 20:43

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To solve

$$ax^2+bx+c=y^2$$

we first multiply both sides by $4a$ to obtain

$$4a^2x^2+4abx+4ac=4ay^2$$

In other words,

$$(2ax+b)^2-(4a)y^2=(b^2-4ac)$$

This is a generalised Pell's equation, which is very nontrivial to solve, e.g http://math.stanford.edu/~conrad/154Page/handouts/genpell.pdf

Because this generalised Pell's equation is equivalent to the original, there's no better way to solve the equation for general $a,b,c$. Of course, in specific instances it is easy to prove that no solutions exist (mods, bounding or otherwise).

auscrypt
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    Thanks for the answer. I know that it leads to a Pell's equation, but I just want to know the solubility or otherwise for specific instances like the example mentioned above ($14144x^2+3872x+265$). I do not want to go into Pell's equation territory since my whole question is as a result of dealing with them. – Wilbur May 29 '19 at 20:26
  • @WilburWestwood but unfortunately, because the general question you asked is exactly the same as a generalised Pell's equation, they're inextricable. For specific instances Pell's equation is sometimes avoidable, but each would have to be dealt with on a case-by-case basis. – auscrypt May 29 '19 at 20:28
  • @WilburWestwood Note that by Theorem 2.1 determining existence of solutions is algorithmic. The theorem may imply exploring a large rectangle of points, though. – logarithm May 29 '19 at 20:30
  • @auscrypt I understand the general case will be inextricably linked to the generalized Pell's equation, but for the specific cases like the one I have mentioned, does modular arithmetic give an indication of the solubility apart from saying it should be 1 mod 8? If yes, how shall I proceed? – Wilbur May 29 '19 at 20:34
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    @WilburWestwood: Your quadratic ($14144x^2+3872x+265$) assumes square values, mod $p$, for all primes $p<10^6$. Based on that, I doubt that modular arithmetic will suffice to resolve the problem. – quasi May 29 '19 at 20:40
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    @WilburWestwood: Do you know for sure that $14144x^2+3872x+265$ is never a perfect square? If so, how do you know? If not, what is the source of this particular example? – quasi May 29 '19 at 20:44
  • I have checked using CAS for x till 10^6.. No perfect square has turned up so far. – Wilbur May 29 '19 at 20:46
  • @WilburWestwood: Where does this particular example come from? Is it an exercise from a book? – quasi May 29 '19 at 20:48
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    @quasi No, it is from a competitive exam question paper from 1979 and I dont have any solution paper for it unfortunately :-( – Wilbur May 29 '19 at 21:09
  • @WilburWestwood: Thanks for the context (although I think you should have disclosed that info as part of the posted question). Can you tell us which contest? – quasi May 29 '19 at 21:15
  • It was not a contest. It was a part of an MPhil (Diploma de Estudios Avanzados) examination question paper from some University in Spain. I was just trawling through some old math question papers and I found it interesting. – Wilbur May 29 '19 at 21:19
  • @WilburWestwood: It is interesting. Based on the source of the problem, do you believe that the intended solution is elementary (e.g., no Pell's equation methods, and no Algebraic Number Theory)? – quasi May 29 '19 at 21:23
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    Well, the question was framed as "14144x^2+3872x+265 can be manipulated to resemble a Pell's equation. Prove that it is not a perfect square for any x?". I can try to make it resemble a negative Pell's equation and see if it has odd period length. But I was wondering what is the point if they give such obvious hint in the question? Plus it was mentioned as a question that requires detailed explanatory answer. – Wilbur May 29 '19 at 21:33
  • Okay so I gave this another shot and hit it with everything I could -- mods, factorisations, sums of squares, cyclotomic polynomials, and everything kinda just bounced off. Not sure if that bodes well for other elementary techniques. – auscrypt May 29 '19 at 21:38
  • @WillJagy i mean, that is literally just pell's equation, which was specifically what the OP asked to avoid in the comment chain – auscrypt May 29 '19 at 21:47