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!)
I spend quite a lot of my time trying to share ideas like this with anyone who will stand still for long enough to listen, because I have a strong sense that many people have not had a taste of what mathematics really is, and that underpins their feeling that they’re not interested in maths, or maths is not for them. My firm belief is that humans have an innate interest in mathematics, but sometimes the way in which ideas are presented can be off-putting. Mathematicians (like everyone else, in fact) spend their time seeking patterns, making predictions and then trying to prove or disprove those, trying to generalise and abstract so as to move from special cases to more general situations, making connections between apparently disparate ideas, and always looking to understand the underlying structures. Mathematics can be beautiful, and doing mathematics requires creativity. The Twin Primes Conjecture exemplifies all this, but there are examples throughout the whole of mathematics (including areas with more direct practical relevance than twin primes, but I’m a number theorist and I love the primes!).
So when I find an audience willing to engage, I try to share those experiences with them, whether in a public lecture to several hundred adults and teenagers, or a workshop in a school with a handful of students. I want them to know what it is to work mathematically, so that they can appreciate mathematical ideas, and perhaps decide to investigate some of them in more depth.
You see, really everyone needs to spend more of their time doing mathematics, it’s just that some of them don’t know it yet…
Dr Vicky Neale
Senior Teaching Associate in the Department of Pure Mathematics and Mathematical Statistics, and Fellow and Director of Studies in Maths at Murray Edwards College