The Music of the Primes: Why an unsolved problem in mathematics matters. Marcus Sautoy du

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СКАЧАТЬ mathematical climax to the twentieth century. The century had opened with Hilbert’s direct challenge to the world’s mathematicians to crack this enigma. Of the twenty-three problems on Hilbert’s list, the Riemann Hypothesis was the only problem to make it into the new century unvanquished.

      On May 24, 2000, to mark the 100th anniversary of Hilbert’s challenge, mathematicians and the press gathered in the Collège de France in Paris to hear the announcement of a fresh set of seven problems to challenge the mathematical community for the new millennium. They were proposed by a small group of the world’s finest mathematicians, including Andrew Wiles and Alain Connes. The seven problems were new except for one that had appeared on Hilbert’s list: the Riemann Hypothesis. In obeisance to the capitalist ideals that shaped the twentieth century, these challenges come with some extra spice. The Riemann Hypothesis and the other six problems now have a price tag of one million dollars apiece. Incentive indeed for Bombieri’s fictional young physicist – if glory weren’t enough.

      The idea for the Millennium Problems was the brainchild of Landon T. Clay, a Boston businessman who made his money in trading mutual funds on a buoyant stock market. Despite dropping out of mathematics at Harvard he has a passion for the subject, a passion he wants to share. He realises that money is not the motivating force for mathematicians: ‘It’s the desire for truth and the response to the beauty and power and elegance of mathematics that drive mathematicians.’ But Clay is not naive, and as a businessman he knows how a million dollars might inspire another Andrew Wiles to join the chase for the solutions of these great unsolved problems. Indeed, the Clay Mathematics Institute’s website, where the Millennium Problems were posted, was so overwhelmed by hits the day after the announcement that it collapsed under the strain.

      The seven Millennium Problems are different in spirit to the twenty-three problems chosen a century before. Hilbert had set a new agenda for mathematicians in the twentieth century. Many of his problems were original and encouraged a significant shift in attitudes towards the subject. Rather than focusing on the particular, like Fermat’s Last Theorem, Hilbert’s twenty-three problems inspired the community to think more conceptually. Instead of picking over individual rocks in the mathematical landscape, Hilbert offered mathematicians the chance of a balloon flight high above their subject to encourage them to understand the overarching lay of the land. This new approach owes a lot to Riemann, who fifty years before had begun this revolutionary shift from mathematics as a subject of formulas and equations to one of ideas and abstract theory.

      The choice of the seven problems for the new millennium was more conservative. They are the Turners in the mathematical gallery of problems, whereas Hilbert’s questions were a more modernist, avant-garde collection. The conservatism of the new problems was partly because their solutions were expected to be sufficiently clear cut for their solvers to be awarded the million-dollar prize. The Millennium Problems are questions that mathematicians have known about for some decades, and in the case of the Riemann Hypothesis, over a century. They are a classic selection.

      Clay’s seven million dollars is not the first time that money has been offered for solutions to mathematical problems. In 1997 Wiles picked up 75,000 Deutschmarks for his proof of Fermat’s Last Theorem, thanks to a prize offered in 1908 by Paul Wolfskehl. The story of the Wolfskehl Prize is what had brought Fermat to Wiles’s attention at the impressionable age of ten. Clay believes that if he can do the same for the Riemann Hypothesis, it will be a million dollars well spent. More recently, two publishing houses, Faber & Faber in the UK and Bloomsbury in the USA, offered a million dollars for a proof of Goldbach’s Conjecture as a publicity stunt to launch their publication of Apostolos Doxiadis’s novel Uncle Petros and Goldbach’s Conjecture. To earn the money you had to explain why every even number can be written as the sum of two prime numbers. However, the publishers didn’t give you much time to crack it. The solution had to be submitted before midnight on March 15, 2002, and was bizarrely open only to US and UK residents.

      Clay believes that mathematicians receive little reward or recognition for their labours. For example, there is no Nobel Prize for Mathematics that they can aspire to. Instead, the award of a Fields Medal is considered the ultimate prize in the mathematical world. In contrast to Nobel prizes, which tend to be awarded to scientists at the end of their careers for achievements long past, Fields Medals are restricted to mathematicians below the age of forty. This is not because of the generally held belief that mathematicians burn out at an early age. John Fields, who conceived of the idea and provided funds for the prize, wanted its award to spur on the most promising mathematicians to even greater achievements. The medals are awarded every four years on the occasion of the International Congress of Mathematicians. The first ones were awarded in Oslo in 1936.

      The age limit is strictly adhered to. Despite Andrew Wiles’s extraordinary achievement in proving Fermat’s Last Theorem, the Fields Medal committee weren’t able to award him a medal at the Congress in Berlin in 1998, the first opportunity after the final proof was accepted, for he was born in 1953. They did have a special medal struck to honour Wiles’s achievement. But it still does not compare to being a member of the illustrious club of Fields Medal winners. The recipients include many of the key players in our drama: Enrico Bombieri, Alain Connes, Atle Selberg, Paul Cohen, Alexandre Grothendieck, Alan Baker, Pierre Deligne. Those names account for nearly a fifth of the medals ever awarded.

      But it is not for the money that mathematicians aspire to these medals. In contrast to the big bucks behind the Nobel prizes, the purse that accompanies a Fields Medal contains a modest 15,000 Canadian dollars. So Clay’s millions will help compete with the monetary kudos of the Nobel prizes. In contrast to Fields Medals and the Faber—Bloomsbury Goldbach prize, the money is there regardless of age or nationality, and with no time limits for a solution, except for the ticking clock of inflation.

      However, the greatest incentive for the mathematician chasing one of the Millennium problems is not the monetary reward but the intoxicating prospect of the immortality that mathematics can bestow. Solving one of Clay’s problems may earn you a million dollars, but that is nothing compared with carving your name on civilisation’s intellectual map. The Riemann Hypothesis, Fermat’s Last Theorem, Goldbach’s Conjecture, Hilbert space, the Ramanujan tau function, Euclid’s algorithm, the Hardy—Littlewood Circle Method, Fourier series, Gödel numbering, a Siegel zero, the Selberg trace formula, the sieve of Eratosthenes, Mersenne primes, the Euler product, Gaussian integers – these discoveries have all immortalised the mathematicians who have been responsible for unearthing these treasures in our exploration of the primes. Those names will live on long after we have forgotten the likes of Aeschylus, Goethe and Shakespeare. As G.H. Hardy explained, ‘languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.’

      Those mathematicians who have laboured long and hard on this epic journey to understand the primes are more than just names set in mathematical stone. The twists and turns that the story of the primes has taken are the products of real lives, of a dramatis personae rich and varied. Historical figures from the French revolution and friends of Napoleon give way to modern-day magicians and Internet entrepreneurs. The stories of a clerk from India, a French spy spared execution and a Jewish Hungarian fleeing the persecution of Nazi Germany are bound together by an obsession with the primes. All these characters bring a unique perspective in their attempt to add their name to the mathematical roll call. The primes have united mathematicians across many national boundaries: China, France, Greece, America, Norway, Australia, Russia, India and Germany are just a few of the countries from which have come prominent members of the nomadic tribe of mathematicians. Every four years they converge to tell the stories of their travels at an International Congress.

      It is not only the desire to leave a footprint in the past which motivates the mathematician. Just as Hilbert dared to look forward into the unknown, a proof of the Riemann Hypothesis would be the start of a new journey. When Wiles addressed the press conference at the announcement of the Clay prizes he was keen to stress that the problems are not the final destination:

      There СКАЧАТЬ