Question

Question #77a3e

Answer 1

Because it would break the Fundamental Theorem of Arithmetic, along with several other important theorems.

Explanation:

Historically, `1` was considered a prime number, but it kept on causing problems, and it was very common for theorems to hold true for "all prime numbers except one".

One very important theorem that wouldn't work if one was considered a prime number is the Fundamental Theorem of Arithmetic. It states that all integers greater than `1` can be represented as a unique product of primes.

The key word there is unique. Let's take `12` as an example integer. Its prime factorization is:
`12=3*2^2`

Suppose you were to allow `1` as a prime number, then you could add `*1` as many times as you wanted to, and it'd still be a product of primes:
`12*3*2^2*1*1...`

This would break the fundamental theorem of arithmetic's property that the prime factorizations are unique. After all, the products would be hardly unique if there were infinitely many.

These types of problems is what eventually led to the exclusion of one as a prime number, because it really only caused problems to the set of prime numbers.