2012 Putnam B6

Let $p$ be an odd prime number such that $p \equiv 2 \pmod{3}$. Define a permutation $\pi$ of the

residue classes modulo $p$ by $\pi(x) \equiv x^3 \pmod{p}$. Show that $\pi$ is an even permutation

if and only if $p \equiv 3 \pmod{4}$.

令 $p$ 为奇素数，使得 $p \equiv 2 \pmod{3}$。定义一个排列 $\pi$

残差类以 $p$ 为模 $\pi(x) \equiv x^3 \pmod{p}$。证明 $\pi$ 是偶排列

当且仅当 $p \equiv 3 \pmod{4}$。