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Most significantly, a proof of the Riemann Hypothesis would mean that mathematicians could use a very fast procedure guaranteed to locate a prime number with, say, a hundred digits or any other number of digits you care to choose. You might legitimately ask, ‘So what?’ Unless you are a mathematician such a result looks unlikely to have a major impact on your life.
Finding hundred-digit primes sounds as pointless as counting angels on a pinhead. Although most people recognise that mathematics underlies the construction of an aeroplane or the development of electronics technology, few would expect the esoteric world of prime numbers to have much impact on their lives. Indeed, even in the 1940s G.H. Hardy was of the same mind: ‘both Gauss and lesser mathematicians may be justified in rejoicing that here is one science [number theory] at any rate whose very remoteness from ordinary human activities should keep it gentle and clean’.
But a more recent turn of events has seen prime numbers take centre stage in the rough and dirty world of commerce. No longer are prime numbers confined to the mathematical citadel. In the 1970s, three scientists, Ron Rivest, Adi Shamir and Leonard Adleman, turned the pursuit of prime numbers from a casual game played in the ivory towers of academia into a serious business application. By exploiting a discovery made by Pierre de Fermat in the seventeenth century, these three found a way to use the primes to protect our credit card numbers as they travel through the electronic shopping malls of the global marketplace. When the idea was first proposed in the 1970s, no one had any idea how big e-business would turn out to be. But today, without the power of prime numbers there is no way this business could exist. Every time you place an order on a website, your computer is using the security provided by the existence of prime numbers with a hundred digits. The system is called RSA after its three inventors. So far, over a million primes have already been put to use to protect the world of electronic commerce.
Every business trading on the Internet therefore depends on prime numbers with a hundred digits to keep their business transactions secure. The expanding role of the Internet will ultimately lead to each of us being uniquely identified by our very own prime numbers. Suddenly there is a commercial interest in knowing how a proof of the Riemann Hypothesis might help in understanding how primes are distributed throughout the universe of numbers.
The extraordinary thing is that although the construction of this code depends on discoveries about primes made by Fermat over three hundred years ago, to break this code depends on a problem that we still can’t answer. The security of RSA depends on our inability to answer basic questions about prime numbers. Mathematicians know enough about the primes to build these Internet codes, but not enough to break them. We can understand one half of the equation but not the other. The more we demystify the primes, however, the less secure these Internet codes are becoming. These numbers are the keys to the locks that protect the world’s electronic secrets. This is why companies such as AT&T and Hewlett-Packard are ploughing money into endeavours to understand the subtleties of prime numbers and the Riemann Hypothesis. The insights gained could help to break these prime number codes, and all companies with an Internet presence want to be the first to know when their codes become insecure. And this is the reason why number theory and business have become such strange bedfellows. Business and security agencies are keeping a watchful eye on the blackboards of the pure mathematicians.
So it wasn’t only the mathematicians who were getting excited about Bombieri’s announcement. Was this solution of the Riemann Hypothesis going to cause a meltdown of e-business? Agents from the NSA, the US National Security Agency, were dispatched to Princeton to find out. But as mathematicians and security agents made their way to New Jersey, a number of people began to smell something fishy in Bombieri’s email. Fundamental particles have been given some crazy names – gluons, cascade hyperons, charmed mesons, quarks, the last of these courtesy of James Joyce’s Finnegans Wake. But ‘morons’? Surely not! Bombieri has an unrivalled reputation for appreciating the ins and outs of the Riemann Hypothesis, but those who know him personally are also aware of his wicked sense of humour.
Fermat’s Last Theorem had fallen foul of an April Fool prank that emerged just after a gap had appeared in the first proof that Andrew Wiles had proposed in Cambridge. With Bombieri’s email, the mathematical community had been duped again. Eager to relive the buzz of seeing Fermat proved, they had grabbed the bait that Bombieri had thrown at them. And the delights of forwarding email meant that the first of April had disappeared from the original source as it rapidly disseminated. This, combined with the fact that the email was read in countries with no concept of April Fool’s Day, made the prank far more successful than Bombieri could have imagined. He finally had to own up that his email was a joke. As the twenty-first century approached, we were still completely in the dark as to the nature of the most fundamental numbers in mathematics. It was the primes that had the last laugh.
Why had mathematicians been so gullible that they believed Bombieri? It’s not as though they give up their trophies lightly. The stringent tests that mathematicians require to be passed before a result can be declared proven far exceed those deemed sufficient in other subjects. As Wiles realised when a gap appeared in his first proof of Fermat’s Last Theorem, completing 99 per cent of the jigsaw is not enough: it would be the person who put in the last piece who would be remembered. And the last piece can often remain hidden for years.
The search for the secret source that fed the primes had been going on for over two millennia. The yearning for this elixir had made mathematicians all too susceptible to Bombieri’s ruse. For years, many had simply been too frightened to go anywhere near this notoriously difficult problem. But it was striking how, as the century drew to a close, more and more mathematicians were prepared to talk about attacking it. The proof of Fermat’s Last Theorem only helped to fuel the expectation that great problems could be solved.
Mathematicians had enjoyed the attention that Wiles’s solution to Fermat had brought them as mathematicians. This feeling undoubtedly contributed to the desire to believe Bombieri. Suddenly, Andrew Wiles was being asked to model chinos for Gap. It felt good. It felt almost sexy to be a mathematician. Mathematicians spend so much time in a world that fills them with excitement and pleasure. Yet it is a pleasure they rarely have the opportunity to share with the rest of the world. Here was a chance to flaunt a trophy, to show off the treasures that their long, lonely journeys had uncovered.
A proof of the Riemann Hypothesis would have been a fitting 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 is a whole new world of mathematics out there, waiting to be discovered. Imagine if you will, the Europeans in 1600. They know that across the Atlantic there is a New World. How would they have assigned prizes to aid in the discovery and development of the United States? Not a prize for inventing the airplane, not a prize for inventing the computer, not a prize for founding Chicago, not a prize for machines that would harvest areas of wheat. These things have become a part of America, but such things could not have been imagined in 1600. No, they would have given a prize for solving such problems as the problem of longitude.
The Riemann Hypothesis is the longitude of mathematics. A solution to the Riemann Hypothesis offers the prospect of charting the misty waters of the vast ocean of numbers. It represents just a beginning in our understanding of Nature’s numbers. If we can only find the secret of how to navigate the primes, who knows what else lies out there, waiting for us to discover?
CHAPTER TWO
The Atoms of Arithmetic
When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question? Enrico Bombieri, ‘Prime Territory’ in The Sciences
Two centuries before Bombieri’s April Fool had teased the mathematical world, equally exciting news was being trumpeted from Palermo by another Italian, Giuseppe Piazzi. From his observatory Piazzi had detected a new planet that orbited the Sun somewhere between the orbits of Mars and Jupiter. Christened Ceres, it was much smaller than the seven major planets then known, but its discovery on January 1, 1801, was regarded by everyone as a great omen for the future of science in the new century.
Excitement turned to despair a few weeks later as the small planet disappeared from view as its orbit took it around the other side of the Sun, where its feeble light was drowned out by the Sun’s glare. It was now lost to the night sky, hidden once again amongst the plethora of stars in the firmament. Nineteenth-century astronomers lacked the mathematical tools for calculating its complete path from the short trajectory they had been able to track during the first few weeks of the new century. It seemed that they had lost the planet and had no way of predicting where it would next appear.
However, nearly a year after Piazzi’s planet had vanished, a twenty-four-year-old German from Brunswick announced that he knew where astronomers should find the missing object. With no alternative prediction to hand, astronomers aimed their telescopes at the region of the night sky to which the young man had pointed. As if by magic, there it was. This unprecedented astronomical prediction was not, however, the mysterious magic of an astrologer. The path of Ceres had been worked out by a mathematician who had found patterns where others had only seen a tiny, unpredictable planet. Carl Friedrich Gauss had taken the minimal data that had been recorded for the planet’s path and applied a new method he had recently developed to estimate where Ceres could be found at any future date.
The discovery of Ceres’ path made Gauss an overnight star within the scientific community. His achievement was a symbol of the predictive power of mathematics in the burgeoning scientific age of the early nineteenth century. Whereas the astronomers had discovered the planet by chance, it was a mathematician who had brought to bear the necessary analytic skills to explain what was going to happen next.
Although Gauss’s name was new to the astronomical fraternity, he had already made his mark as a formidable new voice in the mathematical world. He had successfully plotted the trajectory of Ceres, but his real passion was for finding patterns in the world of numbers. For Gauss, the universe of numbers presented the ultimate challenge: to find structure and order where others could only see chaos. ‘Child prodigy’ and ‘mathematical genius’ are titles that are bandied about far too often, but there are few mathematicians who would argue with these labels being attached to Gauss. The sheer number of new ideas and discoveries that he produced before he was even twenty-five seems to defy explanation.
Gauss was born into a labourer’s family in Brunswick, Germany, in 1777. At the age of three he was correcting his father’s arithmetic. At the age of nineteen, his discovery of a beautiful geometric construction of a 17-sided shape convinced him that he should dedicate his life to mathematics. Before Gauss, the Greeks had shown how to use a compass and straight edge to construct a perfect pentagon. No one since had been able to show how to use this simple equipment to construct other perfect, so-called regular polygons with a prime number of sides. The excitement that Gauss experienced when he found a way to build this perfect 17-sided shape prompted him to start a mathematical diary which he would keep for the next eighteen years. This diary, which remained in the family’s hands until 1898, has become one of the most important documents in the history of mathematics, not least because it confirmed that Gauss had proved, but failed to publish, many results that it took other mathematicians well into the nineteenth century to rediscover.
One of Gauss’s greatest early contributions was the invention of the clock calculator. This was an idea, rather than a physical machine, that unleashed the possibility of doing arithmetic with numbers that had previously been considered too unwieldy. The clock calculator works on exactly the same principle as a conventional clock. If your clock says it’s 9 o’clock, and you add 4 hours, the hour hand moves round to 1 o’clock. Gauss’s clock calculator would therefore return the answer 1 rather than 13. If Gauss wants to do a more complicated calculation such as 7 × 7, the clock calculator would come up with the remainder that is left after dividing 49 = 7 × 7 by 12. The result would again be 1 o’clock.
It is when Gauss wants to calculate the value of 7 × 7 × 7 that the power and speed of the clock calculator begins to emerge. Instead of multiplying 49 by 7 again, Gauss can just multiply the last answer (which was 1) by 7 to get the answer 7. So without having to calculate what 7 × 7 × 7 was (which happens to be 343), he still knew with little effort that it gave remainder 7 on division by 12. The power of the calculator came into its own when Gauss started exploring big numbers that lay beyond his computational reach. Although he had no idea what 799 was, his clock calculator told him that the number gave remainder 7 on division by 12.
Gauss saw that there was nothing special about clocks with 12 hours on their face. He introduced the idea of doing clock arithmetic, sometimes called modular arithmetic, with any number of hours on the clock face. So, for example, if you enter 11 into a clock calculator divided into 4 hours, the answer is 3 o’clock since 11 leaves remainder 3 on division by 4. Gauss’s account of this new sort of arithmetic revolutionised mathematics at the turn of the nineteenth century. Just as the telescope had allowed astronomers to see new worlds, the development of the clock calculator helped mathematicians to discover in the universe of numbers new patterns which had been hidden from view for generations. Even today, Gauss’s clocks are central to the security of the Internet, which utilises calculators whose clock faces bear more hours than there are atoms in the observable universe.
Gauss, the child of a poor family, was lucky to get the chance to capitalise on his mathematical talent. He was born into an age when mathematics was still a privileged pursuit funded by noble courts and patrons, or practised by amateurs such as Pierre de Fermat in their spare time. Gauss’s patron was the Duke of Brunswick, Carl Wilhelm Ferdinand. Ferdinand’s family had always supported the culture and economy of their dukedom. Indeed, his father had founded the Collegium Carolinum, one of the oldest technical universities in Germany. Ferdinand was imbued with his father’s ethos that education was the foundation of Brunswick’s commercial successes, and he was always on the lookout for talent deserving of support. Ferdinand first came across Gauss in 1791, and was so impressed with his abilities that he offered to finance the young man to attend the Collegium Carolinum so that he could realise his obvious potential.
It was with much gratitude that Gauss dedicated his first book to the duke in 1801. This book, entitled Disquisitiones Arithmeticae, collected together many of Gauss’s discoveries about the properties of numbers that he had recorded in his diaries. It is generally acknowledged as the book that heralded the birth of number theory as a subject in its own right, not just a ragbag collection of observations about numbers. Its publication is responsible for making the subject of number theory, as Gauss always liked to call it, ‘the Queen of Mathematics’. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians.
The first tentative evidence that humankind knew about the special qualities of prime numbers is a bone that dates from 6500 BC. Called the Ishango bone, it was discovered in 1960 in the mountains of central equatorial Africa. Marked on it are three columns containing four groups of notches. In one of the columns we find 11, 13, 17 and 19 notches, a list of all the primes between 10 and 20. The other columns do seem to be of a mathematical nature. It is unclear whether this bone, housed in Belgium’s Royal Institute for Natural Sciences in Brussels, truly represents our ancestors’ first attempts to understand the primes or whether the carvings are a random selection of numbers which just happen to be prime. Nevertheless, this ancient bone is perhaps intriguing and tantalising evidence for the first foray into the theory of prime numbers.
Some believe that the Chinese were the first culture to hear the beating of the prime number drum. They attributed female characteristics to even numbers and male to odd numbers. In addition to this straight divide they also regarded those odd numbers that are not prime, such as 15, as effeminate numbers. There is evidence that by 1000 BC they had evolved a very physical way of understanding what it is, amongst all the numbers, that makes prime numbers special. If you take 15 beans, you can arrange them in a neat rectangular array made up of three rows of five beans. Take 17 beans, though, and the only rectangle you can make is one with a single row of 17 beans. For the Chinese, the primes were macho numbers which resisted any attempt to break them down into a product of smaller numbers.
The ancient Greeks also liked to attribute sexual qualities to numbers, but it was they who first discovered, in the 4th century BC, the primes’ true potency as the building blocks for all numbers. They saw that every number could be constructed by multiplying prime numbers together. Whilst the Greeks mistakenly believed fire, air, water and earth to be the building blocks of matter, they were spot on when it came to identifying the atoms of arithmetic. For many centuries, chemists strove to identify the basic constituents of their subject, and the Greeks’ intuition finally culminated in Dmitri Mendeleev’s Periodic Table, a complete description of the elements of chemistry. In contrast to the Greeks’ head start in identifying the building blocks of arithmetic, mathematicians are still floundering in their attempts to understand their own table of prime numbers.
The librarian of the great ancient Greek research institute in Alexandria was the first person we know of to have produced tables of primes. Like some ancient mathematical Mendeleev, Eratosthenes in the third century BC discovered a reasonably painless procedure for determining which numbers are prime in a list of, say, the first 1,000 numbers. He began by writing out all the numbers from 1 to 1,000. He then took the first prime, 2, and struck off every second number in the list. Since all these numbers were divisible by 2, they weren’t prime. He then moved to the next number that hadn’t been struck off, namely 3. He then stuck off every third number after 3. Since these were all divisible by 3, they weren’t prime either. He kept doing this, just picking up the next number which hadn’t already been struck from the list and striking off all the numbers divisible by the new prime. By this systematic process he produced tables of primes. The procedure was later christened the sieve of Eratosthenes. Each new prime creates a ‘sieve’ which Eratosthenes uses to eliminate non-primes. The size of the sieve changes at each stage, but by the time he reaches 1,000 the only numbers to have made it through all the sieves are prime numbers.
When Gauss was a young boy he was given a present – a book containing a list of the first several thousand prime numbers which had probably been constructed using these ancient number sieves. To Gauss, these numbers just tumbled around randomly. Predicting the elliptical path of Ceres would be difficult enough. But the challenge posed by the primes had more in common with the near-impossible task of analysing the rotation of bodies such as Hyperion, one of Saturn’s satellites, which is shaped like a hamburger. In contrast to the Earth’s Moon, Hyperion is far from gravitationally stable and spins chaotically. Even though the spinning of Hyperion and the orbits of some asteroids are chaotic, at least it is known that their behaviour is determined by the gravitational pull of the Sun and the planets. But for the primes, no one had the faintest idea what was pulling and pushing these numbers around. As he gazed at his table of numbers, Gauss could see no rule that told him how far to jump to find the next prime. Were mathematicians just going to have to accept these numbers as determined by Nature, set like stars in the night sky with no rhyme or reason? Such a position was unacceptable to Gauss. The primary drive for the mathematician’s existence is to find patterns, to discover and explain the rules underlying Nature, to predict what will happen next.
