Question 42
$f(x) = (x + b)^2 + c$, where $b$ and $c$ are integers. If the prime $p$ divides $c$, but $p^2$ does not divide $c$, show that $f(n)$ is not divisible by $p^2$ for any integer $n$. If an odd prime $q$ does not divide $c$, but divides $f(n)$ for some $n$, show that for any $r$, we can find $N$ such that $q^r$ divides $f(N)$.