Let $S$ be the set of positive integers of the form $6k+1$ for some integer $k$.
Find an irreducible prime in $S$ such that $p\mid ab$ for some $a,b\in S$ but, $p\nmid a$ and $p\nmid b$.
So the above question is what I'm trying to answer however I'm going to be completely honest here, I'm in elementary number theory and I've tried to look up the definition for irreducible primes but most of the definitions look very abstract to me. I was wondering if someone can please explain what irreducible primes are. I don't feel like my original question would be hard to answer if I knew the definition of an irreducible prime. Thanks