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"...