0

I'm trying to understand why 1 can't be a prime number. Apparently, the reason is that a prime number must be divisible by exactly two numbers: itself and 1. For the number 1's case, since "itself" and 1 are the same number, 1 is technically divisible only by one number. So, what I'm asking here is, if that is the only reason that 1 can't be a prime number, then why must a prime number be divisible by exactly two numbers? Why did mathematicians tack this rule on rather than making the only rule for a prime number being that it must be divisible only by itself and 1?

Mauro ALLEGRANZA
  • 94,169
Cruel
  • 11
  • Welcome to math.SE! <> Does this answer your question? Why is $1$ not a prime number? – Andrew D. Hwang Sep 04 '22 at 23:11
  • Yes, thank you. – Cruel Sep 04 '22 at 23:17
  • 2
    It can, it perfectly can. It is just a definition. Definitions are not content. They are used to organize it. You could define that $1$ is a prime. The statements of some theorems would need to change, but that's all. – plop Sep 04 '22 at 23:22
  • 2
    It is because you lose a lot of properties, like unique factorization into primes. So $12=2\cdot 2\cdot 3,$ but if we allow $1,$ we also have $12=1\cdot 2\cdot 2\cdot 3$ and $1\cdot 1\cdot 2\cdot 2\cdot 3.$ If we define prime to include $1,$ there are a lot of theorems where we have to say, "for primes other than $1.$" In math, we define words mostly based on what we want to talk about. Turns out, most things we want to say about primes we want to exclude $1.$ – Thomas Andrews Sep 04 '22 at 23:46
  • The interesting case is $0.$ When you get to ideals, you'll see that the ideal associated with $0$ is prime, and the other prime ideals correspond to prime integers, so $0$ is prime, in some sense (though not in your definition.) In other rings, you get prime ideals which are not even associated with single elements in the ring. – Thomas Andrews Sep 04 '22 at 23:51

0 Answers0