eli5-Why is the number 1 not considered a prime number even though there isn’t any number that divides it?

In: 2

Prime numbers are defined as numbers that only have two factors – themselves, and 1. Since 1 only has one factor (itself,) by definition it isn’t prime

There are a number of reasons. Fundamentally you could define primes to include it if you wanted, but it turns out to be mathematically convenient not to because a lot of theorems would have to say something like “…primes except 1”, which is a bad way to define a group of things.

For example, the fundamental theorem of arithmetic says that every number (positive integer) can be factored into a *unique* product of primes. But if you include 1, that’s no longer true, since (say) 6 can be factored as 2*3 or 1*2*3.

The common definition of prime is a little bit incomplete, and when you add the concept of unit everything starts making sense

A unit is a number that is invertible, that is, u is a number such that it exists a number a that satisfies a*u = 1. Then, you define a prime as a number that is NOT a unit, and is only divisible by itself times a unit or a unit

In the natural numbers there is only one unit: 1, and because of it being a unit, it is not considered a prime

If you thought about primes with both positive and negative integers, though, you’d have two units: 1 and -1, and numbers like -2, -5 and -7 would also be primes

A basic philosophy in math is that facts should be such that you don’t have to always refer to exceptions. For example, “any number can be factored uniquely into primes” is a perfectly beautiful fact if you do not include 1 as a prime. However, if you include 1 as a prime you now need to say “any number can be factored uniquely except for multiplication by some number of 1’s.”

This also leads us to question defining primes as numbers that cannot be further divided. In doing so we realize that 1 is quite different from all the other primes, so maybe this is not the best definition of prime! Then we try to figure out a better definition of prime that excludes 1.

The fact that primes cannot be further subdivided then becomes a result of the actual definition of prime, and this property just happens to be shared by the number 1.

This is often how mathematics is done. We define something and learn some cool stuff, then realize our definition isn’t the best choice because the resulting facts are not clean.

Prime numbers may be divided by two different numbers. That is the definition. The number 1 may only be divided by a single number. Since it may not be divided by two numbers, it is not prime.

That is also why 2 is prime. It may be divided by two numbers, but two alone.