Is 1 a prime number?

Ellen Hammatt from Te Kura Mātai Tatauranga—School of Mathematics and Statistics explains.

Is one a prime number? Well, that's going to depend on exactly how you define what a prime number is. As a mathematician, when it comes to actually doing maths, an exact definition is very important. There are 2 definitions of a prime number:

1. a number that only 1 or itself divides it
2. a number that has exactly 2 factors.

You might be thinking that these are the same definition, and in fact, they are for all numbers other than 1. In the first definition, we would consider that 1 is going to be a prime number because 1 divides it and 1 divides it. So, 1 and itself. But it only has one factor. So, it's not a prime number when it comes to the second definition.

We have to decide on one of these definitions, and it's going to depend on what properties we want a prime number to have. Something special about prime numbers is that every number has a unique prime factorisation.

What is a prime factorisation? Let's use the example of 6. We can write it as a product of only prime numbers: 6 = 3 x 2 or 2 x 3. But is 3 x 2 a different prime factorisation than 2 x 3? These things are equal in the sense that they result in the same number, 6. But does the order matter? Because we care about this as an object, as a prime factorisation it's going to depend on what we care about. In this example, all we care about is how many of each prime number is needed. So, one 2 and one 3.

We want these to be the same prime factorisation and this has a unique one. But you can imagine that if we allowed 1 to be a prime number, then we could keep adding 1s, and it could have any number of 1s added to the end. So you wouldn't consider it to be unique anymore.

6 = 2 x 3 x 1 x 1 x 1...

So this is one of the reasons why we wouldn’t want to consider 1 to be a prime number because otherwise, we'd have to state the theorem as any number has a unique prime factorisation of primes greater than 1. So 1 doesn't quite act like a prime number in the way that the other numbers do.

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