Why 1 is not a Prime Number

One of the chapters in Steven Strogatz’s The Joy of x is about prime numbers. Everyone knows what a prime number, so I needn’t get into that. Except, there’s the awkwardness about the number 1. Is 1 a prime or not?

“Given that 1 is divisible only by 1 and itself, it really should be considered prime.”

But maths doesn’t consider 1 as a prime. Why not?

 

For convenience. Huh?

“If 1 (0ne) were allowed in, it would mess up a theorem that we’d like to be true. In other words, we’ve rigged the definition of prime numbers to give us the theorem we want.”

What the hell? Which theorem is this? It’s theorem we’ve all learnt in middle school. But before we get to that theorem, let’s recollect what prime factorization means. It’s breaking into a number into its prime factors e.g. 6 = 2 x 3. Or 40 = 2 x 2 x 2 x 5 etc.

 

Now let’s look at the prime factorization theorem. The theorem says that any number can be factored into primes in a unique way. The part in italics would be violated if we allowed 1 (one) to be a prime. Why? Because then we could break 6 into (2 x 3), and (1 x 2 x 3), and (1 x 1 x 2 x 3)… you get the idea. Not unique.

“Silly, of course, but that’s what we’d be stuck with if 1 were allowed in.”

 

This story reveals something:

“This sordid little tale is instructive: it pulls back the curtain on how maths is sometimes done. The naïve view is that we make our definitions, set them in stone, then deduce whatever theorems happen to follow from them. Not so. That would be too passive. We’re in charge and can alter definitions as we please – especially if a slight tweak leads to a tidier theorem, as it does here.”