Why Isn't One a Prime Number?

The number one is not classified as a prime number.

Prime numbers are defined as whole numbers greater than 1 that have exactly two distinct divisors: 1 and the number itself.

Since a prime number must have two distinct factors, one doesn’t qualify—it has only a single divisor (itself).

For this reason, by mathematical convention, 1 is not considered a prime number.

Note. This distinction is crucial because if 1 were considered prime, it would break several fundamental principles of number theory, including the fundamental theorem of arithmetic.

According to the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization, regardless of the order of the factors.

For example, the number 6 can be factored as:

$$ 6 = 2 \cdot 3 = 3 \cdot 2 $$

Here, the prime factorization is unique.

If 1 were classified as a prime number, this uniqueness would be lost.

That’s because we could multiply any prime factorization by 1 indefinitely, generating infinitely many versions of the same factorization.

For instance, if 1 were considered prime, we could write:

$$ 6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3 = 1 \cdot 1 \cdot 2 \cdot 3 \quad \text{and so on} $$

This would make prime factorizations meaningless, as we could tack on as many 1s as we like.

To preserve the uniqueness of prime factorization, 1 is excluded from the set of prime numbers.

And that’s why.