Let $P(x)$ be a polynomials of degree $N$ with real coefficients such that $P(0),P(2),P(3),\cdots,P(N+1)$ are integers. Find all integer $z$ for which there exist a polynomial $P$ that $P(z)$ is not an integer. Note that we don't refer to $P(1)$. my question: Is there any rules for $z$? ex...

Is it true for any $N\geq 1$, there is a value $f(N)$ such that for any integer $x\geq f(N)$ the integers $x+1,x+2,x+3,\ldots ,x+N$ are always multiplicatively independent (i.e. the relation $(x+1)^{e_1}(x+2)^{e_2}\ldots (x+N)^{e_N}$=1 with the $e_k\in{\mathbb Z}$ is possible only when all those ...