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.

  • $\begingroup$ Tried using Diophantine equation but couldnot proceed $\endgroup$ – shirish Apr 12 at 17:20
  • 6
    $\begingroup$ 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

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 $x \to \pm \infty$?

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.