It's this last problem that the Riemann hypothesis tackles. Riemann showed that the number of primes less than some value was controlled by a special function, now known as the Riemann zeta function. All you need to do is find the values that this function sends to zero – the "zeros" of the function – and Riemann reckoned he knew where they were. Although he could never prove it, he conjectured that all the non-trivial zeros lay on one line in the complex plane, and that claim is what we now call the Riemann hypothesis.
Riemann wasn't a number theorist, and only wrote one paper on the subject in his entire career. But this result, even unproven, was so significant that he's still considered one of the most influential figures in the field.
Generations of mathematicians have attempted, with varying degrees of success, to edge closer to a proof, but the hypothesis has so far remained open. If Michael Atiyah – a prolific mathematician who has received just about every award going – has found a proof, it would have huge ramifications both in and outside the mathematical world.
See, prime numbers aren't just a mathematical curiosity. Every time you send a message to a friend or buy something online – even reading this article right now – you're using prime number theory. Modern encryption depends on the fact that primes are very, very hard to predict, and any result as big as this one could have a huge effect on how we keep our data secure.
Intriguingly, by citing von Neumann, Hirzebruch and Dirac, Atiyah hints that his "simple proof" draws influence from the world of quantum mechanics. There's no doubt we should keep a healthy dose of skepticism until the proof is presented and reviewed. However, if it holds up, we could be about to see the most significant discovery in mathematics for over a quarter of a century.