Here's a better formula, that mathematicians actually use:

Let R be an integral domain (a commutative ring with unity where zero is the only zerodivisor). We say an element x is a unit iff x has a multiplicative inverse in R. We say that a non-unit (this is the important part) element p is prime iff, whenever p divides ab, p divides either a or b.

In the case of Z (the set of integers), for example, 2, 3, -5, 17, and -65537 are primes. Yes, the negatives of primes are also primes. In general, all the associates of a prime are prime (a is an associate of b if a = ub for some unit u).

Anyway, my point... whether or not you call 1 a prime has no implications in mathematics whatsoever. However, mathematics is about definitions, and mathematicians must agree on those definitions if they are to be able to communicate at all. If one author calls 1 a prime and another author does not, only confusion results. Not that this doesn't happen---I have seen a large number of mathematics texts say zero is not a natural number, and an even larger number say it is---but it still befits us to eliminate those incongruities and imprecisions whenever possible.

It happens that the above definition, which explicitly excludes units, is usually the most useful one. Why? Because there are a large number of theorems which hold for primes but not for units. `Prime' was chosen to not include units because saying `prime' in most cases and `prime or unit' in the few others is easier than saying `non-unit prime' in most cases and `prime' in a few.