Are there infinitely many pairs of prime numbers that differ by just 2? (A number is prime if it is divisible only by 1 and itself. For example, 3 and 5 are prime numbers that differ by just 2, as are 11 and 13, and as are 107 and 109.)
No, me neither.
There are all sorts of reasons why it seems likely that there are infinitely many pairs of ‘twin primes’ (prime numbers that differ by just 2), but nobody can prove it. Yet.
One of the things that I love about maths is that such an apparently simple, elegant question can turn out to be so challenging. The ancient Greeks proved, more than two thousand years ago, that there are infinitely many prime numbers, via a beautiful argument. It may well be that they also wondered whether there are infinitely many of these pairs of twin primes, and many mathematicians have worked on this problem, but it turns out that it’s really rather difficult. It’s a very famous problem in maths, known as the Twin Primes Conjecture. (A conjecture is a statement that mathematicians think is true, but cannot yet prove. Gathering lots of evidence, perhaps with the help of a computer, just isn’t enough: we have lots of examples of twin primes, some of them involving very large prime numbers, but that’s not the same as a proof that there are infinitely many twin primes. After all, a computer might just have found the largest pair of twin primes!) …continue reading…