Question 159
Let $P \in \mathbb{Q}[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is "nice" if there exists some positive integer $n$ such that $m = n^{3} + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $\left(n_{k}\right)$ of arbitrary length such that $P\left(n_{k}\right)$ are all nice for $k = 1, 2, 3, \ldots$