## Thursday, 2 June 2016

### You cannot prove a negative

Or so I've heard. But it isn't true. Put it with the myth that all codes can be broken if you use enough computer power. Not true.

But can you prove a negative? Some you can't prove. For example, you can't prove that unicorns don't exist. Maybe somewhere, in a galaxy far far away, there's a frolicking herd of unicorns - we don't know. Maybe even in an unexplored part of a dense jungle in Earth.

But some you can prove, and I'm going to give an example.

To prove: there is no largest prime. A prime is an integer that is divisible by one and by itself, and not any other integer, with no remainder. For example: 2, 3, 5, 7, 11, 13 ...

Proof.

Suppose there is a largest prime, P. Then let's line up all the primes less than P, and let's multiply them all together, and add 1. That gives us a number larger than P. And if you try to divide it by any of the lower primes, you get a remainder of 1, so this new larger number is a prime, which is larger than the number we thought was the largest prime.

So, there cannot be a largest prime.

Quod erat demonstrandum

So, there's an example of a negative that can be proved. So not all negatives are unprovable, only some of them.