1990 OIM Problems/Problem 3

Problem

Let $f(x) = (x + b)^2-c$ , be a polynomial with $b$ and $c$ as integers.

a. If $p$ is a prime number such that $p$ divides $c$ and $p^2$ does not divide $c$ , show that, whatever the integer $n$ is, $p^22$ does not divide $f(n)$ .

b. Let $q$ be a prime number other than 2, that divides $c$ . If $q$ divides $f(n)$ for some integer $n$ , show that for every positive integer $r$ there exists an integer $n'$ such that $q^r$ divides $f(n')$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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1990 OIM Problems/Problem 3

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