>not listed as a prime number Can any of you geniuses explain why?

same reason a brick isn't listed as a building

Unicity of prime number factorization. Next thread.

Numbers wouldn't be made up of a unique product of primes otherwise.

Instead of only having 2 * 3 for 6 you could have 2 * 3 * 1 * 1 * 1 * 1, etc.

You go from numbers having a unique product of primes to them having an infinite number of products of primes if you include 1 as prime.

>>9154517
One of the points of prime numbers is to be able to factorise any number with a finite and unique set of primes. If 1 was considered prime, then you could infinitely factorise any number by multiplying it by an infinite number of 1's.

For example, 45, when factored, becomes 5*3*3. If 1 was prime, it could be factored 5*3*3*1*1*1*1.....with an infinite number of 1's.

>>9154517
A prime number is any number that has two factors: 1, and itself.

1 has only one factor: itself. Not a prime.

>>9154544
>>9154546
backwards logic

>>9154553
plain wrong

The only reason we CHOOSE to exclude 1 from primes is because we want the fundamental theorem of arithmetic to exist (every number can be expressed as a unique product of primes)

>>9154577
"plain wrong" isn't an argument, my dude

>>9154577
Then the fundamental theorem of arithmetic must suck. I wouldn't want a theorem that says "this must be true because I had to change facts for no reason to make it true."

That's not math, that's depressing.

>-1 is not listed a prime number
> [math]\frac{1}{2} + \frac{\sqrt{3}}{2} i[/math] is not listed as a prime number

Can any of you geniuses explain why?

When you mod out by a prime ideal, you should get a field. But fields have two elements.

define """"""""""""number"""""""""""""" first of all you fucking drop out

>>9154669
It's a concept

>>9154673
not good enough

>>9154517
Because it is not above 1?

>>9154517
My guess is, that it's simply because the phrase "all primes EXCEPT 1" would be a lot more common than the phrase "all primes AND 1" is right now.

File: CVdBTpfWIAA5dgZ.png (182KB, 600x367px) Image search: [Google]

182KB, 600x367px

>>9154517
Ask this guy, he likes all numbers

>>9154594
What do you think axioms are?

>>9154517
a) 1 is the multiplicative identity. Calling 1 a prime is as baseless as calling 0, the additive identity, positive or negative (it is both not positive and not negative).

b) All positive integers can be written as a unique product of prime numbers. Including 1 as a prime means that no representation would be unique for an incredibly trivial reason. We define what prime is, so might as well not include 1. This helps us mitigate many theorems in number theory without a silly "non-1 primes clause" written before every proof.

>>9154577
>every number can be expressed as a unique product of primes
>0
>1

>>9154723
It's supposed to be positive integers, which doesn't include 0. 1's unique product of primes is no primes (p1^0 x p2^0 x p3^0 x ...).

>>9154693
You can change the definition to
"A prime number is a postive integer with two distinct factors: one and itself"

Then the factors of 1 being 1 and 1 don't satisfy the condition. So it isn't prime.

>>9154723
I'm gonna venture and say 0 is prime; i.e. asking that it be "prime" is exactly working in an integral domain.

>>9154594
>i hate axioms
then don't study math.

>>9154577
If 1 was called a prime then the fundamental theorem of arithmetic would just say "all primes except for 1" and so would every other proof in the world that uses primes. You're not adding anything substantial you're just creating bad communication.