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Science Moments of 2013 – Matt Parker

70 million is a lot bigger than two, I’m prepared to admit that.

The most amazing discovery of 2013 anywhere – by which I mean in mathematics – was an almost proof: it’s a proof, or a step in the right direction, for the twin prime conjecture. Because for a long time mathematicians have said, ‘OK, there’s infinitely many primes, we’ve got that down, but some of them seem to be very friendly.  Some of them come very close together and so, ah, 29 and 31 for example are only one apart.  So there’s one number between them, but otherwise they’re right there, and we call those twin primes.  And for a long time mathematicians have tried to prove there’s an infinite supply of these prime pairs but we’ve had no idea if that’s true or not.

The closest we ever came was in 1915, a Norweigian mathematician who – I’m gonna mispronounce his name – Brun, managed to prove that, at the very least, even though he didn’t prove that they’re finite, that they’re going to run out, he managed to prove that they’re very, very scarce.  Because there’s infinite prime numbers and if you add all the infinite prime numbers together you get an infinite answer, it just gets bigger and bigger and bigger because there’s always more and more numbers you’re adding.  In fact, if you add the inverses of all the prime numbers, so instead of adding 2+3+5+7 you add a half plus a third plus a fifth and all the way up, you still get an infinite answer, you get a huge…it diverges, you’d say in mathematics…it still becomes massive.  Brun managed to prove that if you take the twin prime pairs, add their inverses, it doesn’t explode off to infinity.  In fact, it approaches an exact value of 1.902 and some other digits.  And that’s called Brun’s constant.  And so we knew that there couldn’t be that many and then in May 2013, a rumour spread on the internet that someone in the US – I can’t pronounce his name – Yitang, Tom as he’s often known, was giving a seminar and he’d come up with a proof that there is a limit to…OK, be very careful here…so, the twin prime conjecture is there will always be another pair of primes that are only two apart.  What he managed to prove is there will always be another pair of primes that are 70 million apart.  And, yes, 70 million is a lot bigger than two, I’m prepared to admit that, mathematicians have worked that out, but it’s a lot smaller than infinity!  Because up until then we had no limit whatsoever.  In fact, his limit was just over 63 million but he thought he may as well round it up to 70 million.  Why not?  And so, for the first time in human history, we had a finite bound in the gaps that you would have in primes.  

And so, in fact, people have been working on the proof ever since.  They started an online project where anyone could go and join in.  At the moment, the boundary’s down to about 5,000.  But it keeps changing.  So I imagine by the time someone is watching this it will have changed again but we don’t know if we’re going to get that proof all the way down to two, it doesn’t look like we’ll get it that close, but this is the first year we’ve ever managed to prove there is a limit to how far apart pairs of primes will be. 

There’s no reason whatsoever [to know such a proof]! Although – fun fact – Google, when they were bidding for a suite of patents, put in a bid for 1.902 etc billion dollars and everyone was like, ‘Where did this come from?’, and they were using Brun’s constant as a random thing.