If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First I tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is even, and exactly one factor on the right is divisible by $3$. It follows that for any $n\in{\mathbb ...