Problem

Source: 2018 China North Mathematical Olympiad Grade 11 Test 1 P2

Tags: number theory



Let $p$ be a prime. We say $p$ is good if and only if for any positive integer $a,b,$ such that $$a\equiv b (\textup{mod}p)\Leftrightarrow a^3\equiv b^3 (\textup{mod}p).$$Prove that (1)There are infinite primes $p$ which are good; (2)There are infinite primes $p$ which are not good.