It's easy to
argue definitions about this. It's also
pointless. 1 is
NOT prime, but not just because a gang of
mathematics teachers decided to define a
prime number as a number with
exactly 2
factors. After all, the same gang could just as well have decided to define a prime number as a number divisible only by 1 and by itself.
So why didn't they? Turns out they had a reason...
The reason mathematicians make definitions is so they can capture important concepts. A good definition is one which lets you state interesting theorems simply. And ensuring 1 is not prime makes all theorems easier to state.
For instance, the Fundamental Theorem of Arithmetic states that every natural number has a unique factorization as a product of prime numbers. If we look at 12, say, then 12=2*2*3, and there's no other way to write it as a product of prime numbers.
But if we accepted 1 as prime, we'd have infinitely many different factorizations!
12 = 2*2*3
12 = 1*2*2*3
12 = 1*1*2*2*3
12 = 1*1*1*2*2*3
...
Sure, we could exclude this case, by stating the
theorem as "every natural number has a unique factorization as a product of prime numbers that are not 1".
But why bother? There's never a situation where it's convenient to consider 1 prime to shorten some definition. And the Fundamental Theorem is important enough that we want to define our concepts to match the objects that appear in it. After all, it is the fundamental theorem -- it contains the concepts we want to talk about. And those concepts are of "primes that aren't 1"...