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What is the most important problem in mathematics?
David Hilbert thought he knew. On August 8, 1900, at the International Congress of Mathematicians in Paris, he announced his definitive 23 problems that would come to dominate mathematical research for the next century.
One hundred years later, paying homage to Hilbert in the best mathematical tradition, the Clay Mathematical Insitute announced the Millennium Prize Problems: seven of the most difficult problems still eluding mathematicians at the turn of the millennium. These problems were so important that solving any one would win a mathematician $1 million, yet even with this incentive, only one has been cracked so far.
A good candidate for the most important problem in math, then, might be something that made both of these landmark lists. And it just so happens there's one that fits the bill: the Riemann hypothesis.
"Ask any professional mathematician what the single most important open problem in the entire field is," wrote mathematician Keith Devlin in 1998, "and you are almost certain to receive the answer 'the Riemann Hypothesis'".
And now, there's a rumor circulating that this 160-year-old mega-problem might finally be solved – by British-Lebanese geometer Sir Michael Atiyah.
First conjectured by Bernhard Riemann in 1859, the Riemann hypothesis is an attempt to answer an age-old question about primes – numbers that divide only by themselves and one. Primes are one of the most fundamental concepts in mathematics – the "atoms" of numbers – but they are also mysterious and inexplicable, turning up seemingly randomly along the number line. That's why it's such big news when mathematicians find new, enormous primes. Even if we find one prime number, there's no way to predict what the next one is or how many there are below it.
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.
