The Prime Debate: Is 1 prime?

[Guest]

0 is not a prime, by my definition above
0 is even. it is also divisble by 3,
In facti it is divisble by every number.

[orange]

i don't think 0 qualifies as a positive number, either

LulThyme - I'm sorry but I still disagree with you. 1 is Prime, in my eyes.

[Peter Triggs]

LulThyme and KrazyHorse are correct. There is nothing to argue about.

A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable."

[connorkimbro]

It is my strong opinion that Enigma only started this thread so that he could demonstrate to everyone how smart he is.

[Guest]

Well orange it depends on the context
usually if you want to exclude, you woudl say Strictly positive, usually positive includes 0 (as does negative btw)
Because 0 is the neutral element with respect to addition, so it makes the positive integers into a monoid. (another technical term)
The thing about math (at least this kind of math) si you cant argue it.
Primes are DEFINED as excluding units, that means 1.
That means that if you ask anybody who actually works with primes, there is no ARGUMENT about if 1 is a prime, you could say there is no reason either, it is just defined that way.
Just like if I DEFINE f(x)= 2x.
You cant argue that its not 2x.
Its the definition.
Now if you want to know why they define 1 as not being a prime, thats another story, with part of it on the first page already, its because it makes proving theorems easier, and makes them nicer (very important in mathematics).

[Guest]

You have to understand that 1 is very special.
If we look at positive integers (thus including 0).
0 is very special because it is the additive neutral element, it is the multiplicative absorber (or whatever in english), and you can't divide anything by it.
Those 3 properties go hand in hand and 0 works like this in every ring.
Now 1 is also special, although only for multiplication (it has no special addition property).
First it is the neutral element.
And second because it is a neutral element it is a unit (although sometimes in other rings there are other units like in the ring of matrices)
This is the property that makes 1 special in our thread.
1 is a unit, so has a multiplicative inverse, so every number can be divided by it, and even more, since it is the neutral element, you get the same number.
So the very important fact is that
1*a=a for all a.
There is this very important theorem called Unique factorization. It means that there is only one way to factorize any strictly positive integer such that none of the factors can be further factorize.
So take any pos integer but 0, say 9.
it can be factorized as 3*3. none of these can be factorized further if we dont allow 1*3 as a factorization and so it is unique.
If we allow it, we can get 1*3*3 or 1*1*3*3 and so on

The key is here : the factorization process does not stop anymore, it is endless.
If 1 is not a prime and we count1*a as trivial, it is obvious that the factorization process will stop and nothing will be able to be further reduced.

[Guest]

I thinkn adding this might help.
If we consider 1 a prime, no computer crashes, no algorithm stop working, no sky falls anywhere.
All math keeps working, we just have to add a lot of exceptions all over the place. Make exceptions in proofs and algorithms for 1. All primes except 1. etc...
So in a way it is an aesthetic choice.
Some of you (orange ) might think that makes you right, but in fact it removes your last argument.
Even if you look at it from a "choice" point of you, the easiest and simplest choice is to exclude 1 from the list of prime, which I think even Euclid understood.

[Ramo]

it is the multiplicative absorber (or whatever in english)

That'd be "fixed point."

[KrazyHorse]

Originally posted by Sava
1 is prime... the definition a prime number is:

a number divisble by only itself and 1... is 1 divisible by any other number other than itself (1) or 1? no...

every number can be infinitely divided by 1, so that argument is meaningless...

EDIT: Damn you orange!

The two people with math degrees here (as I recall LulThyme was a mathie) are both telling you guys the same thing: 1 is not considered prime. It's always mentioned specifically in giving the definition of primes that 1 is not a prime...

[Kuciwalker]

Plus, I'll just add my dad's voice to this (my dad has a masters in mathematics and is an actuary) - one is not a prime.