Dear Frank:

> Somewhere on Chris Caldwell's site, > http://www.utm.edu/research/primes/
> he discussed whether or not the number one was prime.

I doubt there's a discussion. It's a matter of definition and its
motivation.

It's a common convention to not consider 1 a prime. Otherwise,
many definitions and theorem would have to deal with a special case.

By definition, a number is prime if, besides 1, it's only divisible by itself.

In this form, 1 is automatically excluded as it can't be "besides 1."

But regardless, a better definition may be the more explicit:

n > 1 is a prime if it's only divisible by 1 and itself.

The Fundamental Theorem of Arithmetic states that any number can be uniquely represent as a product of primes. If 1 is a prime the representation can't be unique, for 1*2 = 1*1*2 = ...

You have to add that there are products that consist of a single term - primes' decomposition into factors. You may like this or not. For whatever reasons, mathematicians like this situation more than accepting 1 as a prime, but than modifying the FTA.

> if every N is representable as the
> product of 2 primes then
> one must be prime.

This is not a prerequisite that there should be at least 2 factors, and in fact contradicts the definitions.

> P.S. Love your site

Thank you

All the best,
Alexander Bogomolny