To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are

To make matters worse, it turns out that there are actually two different concepts commonly labelled by the word "prime", and in some highbrow contexts they are different -- so mathematicians actually define "prime" in an entirely different way: p is prime iff p|ab => (p|a or p|b). That is, a prime is something that can't divide the product of two numbers without dividing at least one of the numbers. The thing about having no nontrivial divisors is called being "irreducible". (When are they different? When you're working with some kind of number other than the ordinary integers. For instance, suppose you decide to treat sqrt(-5) as an integer, so your numbers are now everything of the form a + b sqrt(-5) where a,b are ordinary integers. In this system of numbers, 3 is irreducible, but it isn't prime because (1+sqrt(-5)) (1-sqrt(-5)) = 6 which is a multiple of 3 -- but neither of those factors is a multiple of 3 even in this extended system of numbers.)