# What is an irreducible prime?

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

I suppose that the answer is something like this: $25,55,115\in S$, $25\mid55\times115$, but $25\nmid55$ and $25\nmid115$. Besides, $25$ is an irreducible prime in $S$, because it can't be written as the product of two smaller elements of $S$.

• Okay so what you're saying is that when we define the set $S$ we give new definition to what is a prime. Prime numbers now only exist within the integers in $S$ correct? – Luis Robles Mar 27 '18 at 7:41
• @LuisRobles Yes, that's it. – José Carlos Santos Mar 27 '18 at 7:44
• Thanks, but then what is the difference between an irreducible and a prime and an irreducible prime? – Luis Robles Mar 27 '18 at 7:46
• @LuisRobles I don't know. That's why I started my answer with “I suppose”. – José Carlos Santos Mar 27 '18 at 7:47
• I think they might be looking for an element which really is a prime (in the integers). But it is hard to tell from the formulation. – Tobias Kildetoft Mar 27 '18 at 7:48