Why is $1$ not a prime number?

Why is $1$ not considered a prime number?

Or, why is the definition of prime numbers given for integers greater than $1$?

Solution 1:

One of the whole "points" of defining primes is to be able to uniquely and finitely prime factorize every natural number.

If 1 was prime, then this would be more or less impossible.

Solution 2:

It's important to understand that this is not something that can be proved: it's a definition. We choose not to regard 1 as a prime number, simply because it makes writing lots of theorems much easier.

Noah gives the best example in his answer: Euclid's theorem that every positive integer can be written uniquely as a product of primes. If 1 is defined to be a prime number, then we'd have to change that theorem to: "every positive integer can be written uniquely as a product of primes, except for infinite multiplications by 1". So we choose to go with the easier path of defining 1 to not be a prime.