The number one is not a prime number because it is the base unit of the number system, and the base unit has special properties that require it to be kept separate from the set of prime numbers. Therefore, prime numbers are defined as integers that are greater than one and that only have one and themselves as divisors.
Prime numbers serve the purpose of defining composite numbers. Composite numbers are all positive integers other than one and prime numbers. Math theorems prove that each composite number can be reduced to a unique multiplication problem of only prime numbers. This is called its prime factorization. For example, the composite number 476 can be reduced to the prime factorization 2 x 2 x 7 x 17. Each number in the multiplication problem is a prime number. If one was a prime number, this would not work. Both 1 x 2 x 2 x 7 x 17 equals 476 and 1 x 1 x 1 x 2 x 2 x 7 x 17 equals 476. The ones make the process confusing and decrease its consistency.
Theorems like this helped mathematicians to realize that one was a special number that needed its own category. One is now classified as a unit along with negative one. Both one and negative one are the only integers that can be multiplied by an integer to produce the number one.