World of maths aflutter over new proof of 160-year-old hypothesis

The 90-year-old mathematician Michael Atiyah has presented what he referred to as a "simple proof" of the Riemann hypothesis, a problem which has eluded mathematicians for almost 160 years.

The world of maths and the Twitter-sphere have been a frenzy ever since the British-Lebanese mathematician indicated he was to give a lecture on Monday at the Heidelberg Laureates Forum in Germany on what is widely regarded as the most important outstanding problem in maths.

Although it is almost incomprehensible for people without intensive maths training, the hypothesis describes the distribution of prime numbers among positive integers.

Prime numbers, very simple by definition, are the building blocks of modern mathematics, especially number theory. Achievements in prime number theory have been widely applied to computer sciences and telecommunications.

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However prime numbers are also mysterious and inexplicable – the pattern in which prime numbers emerge in the line of positive integers has remained elusive to generations of mathematicians.

Riemann proposed a theory that can, in a way, shed light on that mystery; but he could not prove it, nor could all the brilliant minds that came after.

If a solution to the Riemann hypothesis is confirmed, it would be big news as mathematicians would be armed with a map to the location of all such prime numbers; a breakthrough with far-reaching repercussions in the field.

Atiyah, who specialises in geometry, is one of the UK's most eminent mathematical figures, having received the two awards often referred to as the Nobel prizes of mathematics; the Fields Medal and the Abel Prize.

$1 million prize

As one of the six unsolved “Clay Millennium Problems”, any solution would be eligible for a $1 million prize. This has tempted many mathematicians over the years, none of which has yet been awarded the prize.

Atiyah is well aware of this history of failure. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he said, but he hoped his presentation would convince his critics.

In it, he paid tribute to the work of two great 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, whose developments he claims laid the foundations for his own proposed proof. "It fell into my lap, I had to pick it up," he said in advance of giving his talk.

He has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers.

“People say ‘we know mathematicians do all their best work before they’re 40’,” Atiyah said recently. “I’m trying to show them that they’re wrong. That I can do something when I’m 90.”

Irish mathematician Prof Peter Lynch of UCD, who tuned into the talk, said in his view there was very little chance that Atiyah was correct: "Major proofs normally require large advances in understanding, while Atiyah has claimed a 'simple proof'. His outline may lead to a complete proof or, more likely, may amount to nothing."

To win a millennium prize the solution has to be considered valid for a period of two years – though other mathematicians are likely to give their verdict in coming weeks.

The Clay Millennium Prizes, announced in 2000, were conceived to record seven of the most difficult problems with which mathematicians were grappling at the turn of the second millennium.

They were also designed “to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasise the importance of working towards a solution of the deepest, most difficult problems; and to recognise achievement in mathematics of historical magnitude”.

[ Has Michael Atiyah conquered the Everest of mathematics? ]