We define a prime number as a positive integer greater than 1 that has no other dividers besides itself and 1. Thus 2 certainly is prime.

It is not really useful to include 1 as a prime as it divides everything. One of the fundamental problems in mathematics (and in numerical codes) that we deal with is writing other integers in terms of products of primes, that is, in terms of the primes that divide them, for example 60 = 2235; the primes are the building blocks for all integers. Including 1 as prime adds nothing to the description of the number as a product of prime factors.

Another consideration is that there is essentially only one way to build an integer as a product of primes. Sixty is the product of two twos, one three and one five. You can change the order in which you multiply them but you need exactly two twos, one three and one five. If one were a prime then we would lose this uniqueness. You could then write sixty as a product of primes in many different ways; 60 = 12235 or 60 = 112235 or 60 = 1112235 or ,,,