Let $ p$ be a prime number. For every $ n \in \mathbb N$ , let $ A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $ p^n|P(m), \forall m \in \mathbb Z$ $ \}$ ; then how to show that $ A_{p,n}$ is non-empty ? If we define $ f(n,p):=\min A_{p,n}$ , then howRead more