Omni-MATH / Find all integers n≥2 for which there exists an integer m and a polynomial P(x) with integer coefficients satisfying the following three conditions: [list] [*]m>1 and gcd(m,n)=1; […

Find all integers n≥2 for which there exists an integer m and a polynomial P(x) with integer coefficients satisfying the following three conditions: [list] 	[*]m>1 and gcd(m,n)=1; 	[*]the numbers P(0), P²(0), …, P^(m−1)(0) 		are not divisible by n; and 	[*]P^(m)(0) is divisible by n. [/list] Here P^(k) means P applied k times, so P¹(0)=P(0), P²(0)=P(P(0)), etc.

[i]Carl Schildkraut[/i]

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] 	[*]$m > 1$ and $\gcd(m,n) = 1$; 	[*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ 		are not divisible by $n$; and 	[*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc.

[i]Carl Schildkraut[/i]