OIM 1990 Problem 3

Let $b$ , $c$ be integer numbers, and define $f(x)=(x+b)^2-c$ .

i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$ , show that for every integer $n$ , $f(n)$ is not divisible by $p^{2}$ .

ii) Let $q \neq 2$ be a prime divisor of $c$ . If $q$ divides $f(n)$ for some integer $n$ , show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$ .