My theory is that it’s just a matter of convenience. It’s easier to say “for every prime number” rather than always having to say “for every prime number except one.”
Anyone got a better reason?
The most important thing about primes is what’s called the unique prime factorization theorem. Given any positive integer, there is one and only one to express it as a product of primes. So, for instance, 12 = 2*2*3, and 42 = 2*3*7. But if we counted 1 as a prime, then there’d be an infinite number of ways, since 12 is also 1*2*2*3, or 1*1*1*1*2*2*3, etc.
Chronos, that looks interesting, but I’m afraid my math-deficient brain isn’t scanning it. Could you expand a bit for those of us for whom math isn’t intuitive? Why is 12 = 223?
Couldn’t you just say, “Given any positive integer there is one and only one way to express it as a product of primes other than one.”?
I’m sure there’s some munched multiplies in there.
So, for instance, 12 = 2*2*3, and 42 = 2*3*7. But if we counted 1 as a prime, then there’d be an infinite number of ways, since 12 is also 1*2*2*3, or 1*1*1*1*2*2*3, etc.
Yes, we could, but it’s longer and harder to deal with than a simple rule without an exception. Ultimately it is a matter of convenience: a lot of theorems and things would get more complicated if 1 were considered a prime, and very few would get simpler.
Is 1 a positive integer? If so, how do you express it as a product of primes?
Gah, I forgot that asterisks make Discourse go stupid. I’ve fixed my post.
Yes, but since statements like that are the reason we’re interested in primes in the first place, why ever include 1 at all?
1 is the product of no primes. No primes is the only way to express 1, and it’s the only number that’s the product of no primes.
Maybe so that you could give third-graders (or maybe the average American) a simple definition of prime numbers rather than having to explain the exception for one over and over again.
This is what it comes down to. “The primes” as they are currently defined now are a useful set the shows up in a lot of theorems. “The primes and also one” is a less useful set that doesn’t show up in very many theorems.
If you redefine the primes to include one, you end up having to say “primes other than one” a lot more, and don’t get very much savings from being able to say just “the primes” for those few theorems where “primes + 1” is an important set.
“A number that has exactly two factors, 1 and itself”
Like -1 = -1 x 1? 
1 x 1 = 1
How many factors are in that equation?
No. It has infinite factors:
1 x 1
1 x 1 x 1
1 x 1 x 1 ad infinitum
Note that if 1 is prime this is true of everything:
2 x 3 x 1 x 1 x 1 x 1 is still 6.
Surely you meant for any non-prime positive integer…5 is a positive integer, but is not factorable as the product of 2 primes - because it is prime.
When a mathematician refers to a product, it could be the product of as few as 0 factors, the product of no factors being 1. (Think of it as starting with 1 and then multiplying by all of the given factors. If there are none, you just have 1 as the result.) If you have just one factor, that factor is your result. So, prime numbers also have unique expressions as products of primes, with just one factor, being the prime itself.
Yeah. It’s bad enough having to count 2 as a prime - there’s a number of theorems about “every odd prime” (meaning every prime but 2). Adding 1 to the club would (as @Chronos and others have explained) much worse
So you agree that it’s that way just because it’s more convenient. Not that there is anything wrong with that.
It also makes functions like
work.
I’m still not understanding who is inconvenienced by the definition. It’s a technical term; a technical definition is to be expected.
Should math also drop the word exponent because it’s confusing? It also has a non-mathematical definition.
a person who believes in and promotes the truth or benefits of an idea or theory.
“an early exponent of the teachings of Thomas Aquinas”
Well, yes, every definition in math is because it’s convenient. We could do without definitions, and just use the full description instead of the single word every time, but that would be really inconvenient.
There are some theorems about the odd primes, but the ones for which you do want to count 2 are far more numerous and significant. And for that matter, there are some for which you want to exclude both 2 and 3, like “(almost) all primes are of the form 6n+1 or 6n+5”. Or exclude 2, 3, and 5, or whatever.