# What is this “prime” thing?

I suppose before we start, it’s important to define a prime number. It turns out that the definition you’ve probably been using most of your life is wrong. It also turns out that this is irrelevant.

A prime number is a number $p$ such that if $a\cdot b$ is divisible by $p$, then either $a$ or $b$ must be divisible by $p$. Also, $1$ and $-1$ are stupid, so we don’t call them primes. Yes, this is (more or less) the official definition.

Wah? Primes can be negative? Sure can! Also, if you look carefully, you’ll see that zero is prime. This is important in ring theory, but not for us. From now on, we’ll only worry about the positive primes.

Okay, but this isn’t the definition you’ve learned for primes. You’ve probably been using irreducible numbers. These are numbers which are only divisible by $1$ and themselves. We’ll again, only care about positive irreducibles. Oh, and also $1$ is stupid, so it doesn’t count as irreducible.

So what’s the difference? Well, not really anything. Primes and irreducibles are the same, at least for positive integers. If you learn some ring theory, you’ll find out these definitions generalize to any ring, and primes and irreducibles are the same whenever we have unique factorization.

What’s unique factorization? It’s when every number can be factored in essentially one way. As mind-boggling as it is, there are some “number systems” that sadistic mathematicians have made up which fail to have unique factorization.

Okay, enough tomfoolery. I wanted to mention this, because from now on, I’ll be using the word “prime,” with the definition for irreducible. Hopefully you’re fine with this, because you’ve been doing that too. Oh, and also, I’ll be using unique factorization. This is somehow a bigger theorem, and I’ll try to avoid using it if I can, but it’ll definitely come up. I’ll try point out when it does.