The big news last year was the quest to find a lower bound for the gap between pairs of large primes, started by Yitang Zhang and carried on chiefly by Terry Tao and the fresh-faced James Maynard.

Now that progress on the twin prime conjecture has slowed down, they’ve both turned their attentions toward the opposite question: what’s the biggest gap between subsequent small primes?

It turns out that results about prime numbers behave a lot like prime numbers: you can wait ages for one and then you run into two very close together. Indeed, Maynard and Tao (along with Kevin Ford, Ben Green and Sergei Konyagin) within days of each other revealed solutions to a problem of some eighty years’ standing, proving that for any value $c$ there are infinitely many pairs $p_n,p_{n+1}$ satisfying $$p_{n+1}-p_n > c\frac{\log n \log\log n \log\log\log\log n}{(\log\log\log n)^2}$$ (where $p_i$ is the $i$th prime number). Paul Erdős offered $10,000 for a proof of this logtastic fact – his most generous reward, making this quantifiably the most important result in the history of mathematics.

More Information

Large gaps between consecutive prime numbers by Ford, Green, Konyagin and Tao on the ArXiv

Large gaps between primes by Maynard on the ArXiv

Ben Green’s announcement of the result as part of his lecture ‘Approximate algebraic structure’ at the International Congress of Mathematicians (YouTube video)