Why Isn't One Considered a Prime Number?

The number one is not classified as a prime number.

Prime numbers are defined as whole numbers greater than 1, divisible only by 1 and by themselves.

For a number to be prime, it needs to have exactly two distinct divisors: 1 and the number itself.

Since 1 has only one divisor—namely, itself—it doesn’t satisfy this requirement.

Therefore, by mathematical convention, 1 is not considered prime.

Note: This definition is essential because, if 1 were regarded as prime, key properties in number theory, like the Fundamental Theorem of Arithmetic, wouldn’t hold.

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 instance, the number 6 can be expressed as:

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

Here, the factorization is unique.

If we counted 1 as a prime number, this uniqueness would be lost.

In fact, I could multiply any prime factor by 1 as many times as I want, generating endless variations of the same factorization.

For example, if 1 were prime, I 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 mean that a number’s factorization is no longer unique, as I could keep adding factors of 1 indefinitely.

To avoid this ambiguity, 1 is excluded from the set of prime numbers.

And so on.