Is 1 a Prime Number?

The definition it fails

A prime number is a whole number with exactly two distinct factors: 1 and the number itself. Count the factors of 1 and there is one of them. Two means two, and the word distinct is there precisely so that 1 cannot be counted twice as both its own factors.

Notice the third case in the list below. The number 1 is not composite either, because a composite number is one that can be made by multiplying two smaller whole numbers greater than 1, and no such pair exists for 1. Mathematicians call it a unit, and it is the only positive whole number that is neither prime nor composite.

• 7 → factors 1 and 7 → exactly two → prime
• 9 → factors 1, 3 and 9 → three → composite
• 1 → factor 1 → one → neither prime nor composite

The reason mathematicians drew the line there

The definition was chosen to protect one statement: every whole number greater than 1 can be written as a product of primes in exactly one way, apart from the order they are written in. 12 is 2 × 2 × 3 and there is no other prime factorisation of it anywhere.

Let 1 count as prime and that uniqueness dies on the first line. 12 becomes 2 × 2 × 3, and also 1 × 2 × 2 × 3, and also 1 × 1 × 2 × 2 × 3, and so on without end. Every number would have infinitely many prime factorisations, and the theorem would have to be rewritten with an exception clause naming 1 - which is the same exclusion, just moved somewhere less convenient.

That theorem is not an ornament. Reading the highest common factor and lowest common multiple off two prime factorisations depends on it. So does knowing that simplifying a fraction terminates, that a surd has one simplest form, and that the number of factors of 720 can be counted without listing them. Take away uniqueness and all of that becomes guesswork.

It is fair to call this a decision rather than a discovery, and honest to say so out loud. But it is not an arbitrary decision. The alternative costs a theorem that thousands of results rest on, and buys nothing at all in return.

What 1 is, since it is neither

1 is the multiplicative identity: multiplying by it changes nothing. That is exactly what disqualifies it as a building block. Primes are the pieces numbers are built from, and a piece that adds nothing to what it is joined to is not a piece.

It has plenty of other jobs. 1 is a factor of every whole number. It is the highest common factor of any two numbers that share nothing else, which is what coprime means. It is a square number, a cube number and a triangular number all at once. None of those categories has anything to do with being prime, and a number can belong to several of them or to none.

The three questions that arrive with this one

The list below settles the ones that follow immediately after somebody has been told that 1 is not prime.

The last of them points at the practical skill. To test whether a number under 100 is prime, you only need to try dividing by primes up to its square root - 2, 3, 5 and 7, since 10 × 10 = 100. That is four divisions rather than ninety, and it catches 91 straight away on the 7.

• Is 2 prime? Yes. It is the only even prime, because every other even number has 2 as a third factor on top of 1 and itself.
• Is 0 prime? No. Every whole number divides into 0 exactly, so it has infinitely many factors rather than two.
• Is 9 prime? No - 9 = 3 × 3. Along with 51 = 3 × 17 and 91 = 7 × 13, it is one of the numbers most often mistaken for a prime.