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Prove that if $P(x)$ is a polynomial with integer coefficients such that $P(n)$ is a perfect square for every integer $n$, the degree of $P(x)$ must be even.

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  • $\begingroup$ Tried using Diophantine equation but couldnot proceed $\endgroup$ – shirish Apr 12 at 17:20
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    $\begingroup$ If it has odd degree, then it takes negative values. $\endgroup$ – user647486 Apr 12 at 17:23
  • $\begingroup$ instead of pulling apart a perfect-square polynomial, try to consider how it may have been built. $\endgroup$ – John Joy Apr 13 at 14:35
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You don't need any of the machinery of the analysis of Diophantine equations.

Hint What is the behavior of an odd polynomial $P(x)$ as $x \to \pm \infty$?

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  • $\begingroup$ It is negative both ways $\endgroup$ – shirish Apr 12 at 17:29
  • $\begingroup$ When x is positive it is opposite and reverses when x is negative $\endgroup$ – shirish Apr 12 at 17:35
  • $\begingroup$ for even degree it is positive both ways $\endgroup$ – shirish Apr 12 at 17:38
  • $\begingroup$ Square of a number is always positive $\endgroup$ – shirish Apr 12 at 17:39

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