The Music of the
Primes Why an Unsolved Problem in Mathematics Matters

Marcus du Sautoy

Copyright

Fourth Estate

An imprint of HarperCollinsPublishers Ltd. 1 London Bridge Street London SE1 9GF

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This edition published by Harper Perennial 2004

First published by Fourth Estate 2003

Copyright © Marcus du Sautoy 2003

PS section copyright © Josh Lacey 2004

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Marcus du Sautoy asserts the moral right to be identified as the author of this work

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Praise

From the reviews of The Music of the Primes:

‘A gripping, entertaining and thought-provoking book. Du Sautoy is certainly a brilliant storyteller and introduces us to some great personalities … Even if you don’t understand the maths, this is still a fascinating book. And if you do understand some of it, it will have you running for your calculator as you try to work out some of the riddles along the way’

Independent on Sunday

‘Delightfully entertaining … [Du Sautoy] has been successful in setting up a compelling dramatis personae of mathematicians, with every character vividly illuminated with anecdotes and felicitous comment’

Guardian

‘Du Sautoy … laces the ideas with history, anecdote and personalia – an entertaining mix that renders an austere subject palatable … Even those with a mathematical allergy can enjoy du Sautoy’s depictions of his cast of characters’

The Times

‘The subject is daunting, but du Sautoy writes with admirable clarity and verve’

Daily Mail

‘An engaging and accessible history of work on prime numbers and the Riemann hypothesis’

Economist

‘Entertaining … looks certain to be a great success’

Nature

Dedication

For the memory of

Yonathan du Sautoy October 21, 2000

Contents

Cover

Title Page

Copyright

Praise

Dedication

1 Who Wants To Be a Millionaire?

2 The Atoms of Arithmetic

3 Riemann’s Imaginary Mathematical Looking-Glass

4 The Riemann Hypothesis: From Random Primes to Orderly Zeros

5 The Mathematical Relay Race: Realising Riemann’s Revolution

6 Ramanujan, the Mathematical Mystic

7 Mathematical Exodus: From Göttingen to Princeton

8 Machines of the Mind

9 The Computer Age: From the Mind to the Desktop

10 Cracking Numbers and Codes

11 From Orderly Zeros to Quantum Chaos

12 The Missing Piece of the Jigsaw

Keep Reading

Acknowledgements

Further Reading

Illustration and Text Credits

Index

P.S.

About the Author

Portrait of Marcus du Sautoy

Snapshot

Top Ten Favourite Books

About the Book

A Critical Eye

Jerzy Grotowski

Read On

If You Loved This, You’ll Like …

Find Out More

About the Author

About the Publisher

CHAPTER ONE
Who Wants To Be a Millionaire?

‘Do we know what the sequence of numbers is? Okay, here, we can do it in our heads … fifty-nine, sixty-one, sixty-seven … seventy-one … Aren’t these all prime numbers?’ A little buzz of excitement circulated through the control room. Ellie’s own face momentarily revealed a flutter of something deeply felt, but this was quickly replaced by a sobriety, a fear of being carried away, an apprehension about appearing foolish, Unscientific. Carl Sagan, Contact

One hot and humid morning in August 1900, David Hilbert of the University of Göttingen addressed the International Congress of Mathematicians in a packed lecture hall at the Sorbonne, Paris. Already recognised as one of the greatest mathematicians of the age, Hilbert had prepared a daring lecture. He was going to talk about what was unknown rather than what had already been proved. This went against all the accepted conventions, and the audience could hear the nervousness in Hilbert’s voice as he began to lay out his vision for the future of mathematics. ‘Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?’ To herald the new century, Hilbert challenged the audience with a list of twenty-three problems that he believed should set the course for the mathematical explorers of the twentieth century.

The ensuing decades saw many of the problems answered, and those who discovered the solutions make up an illustrious band of mathematicians known as ‘the honours class’. It includes the likes of Kurt Gödel and Henri Poincaré, along with many other pioneers whose ideas have transformed the mathematical landscape. But there was one problem, the eighth on Hilbert’s list, which looked as if it would survive the century without a champion: the Riemann Hypothesis.

Of all the challenges that Hilbert had set, the eighth had a special place in his heart. There is a German myth about Frederick Barbarossa, a much-loved German emperor who died during the Third Crusade. A legend grew that he was still alive, asleep in a cavern in the Kyffhäuser Mountains. He would awake only when Germany needed him. Somebody allegedly asked Hilbert, ‘If you were to be revived like Barbarossa, after five hundred years, what would you do?’ His reply: ‘I would ask, “Has someone proved the Riemann Hypothesis?”’

As the twentieth century drew to a close, most mathematicians had resigned themselves to the fact that this jewel amongst all of Hilbert’s problems was not only likely to outlive the century, but might still be unanswered when Hilbert awoke from his five-hundred-year slumber. He had stunned the first International Congress of the twentieth century with his revolutionary lecture full of the unknown. However, there turned out to be a surprise in store for those mathematicians who were planning to attend the last Congress of the century.

On April 7, 1997, computer screens across the mathematical world flashed up some extraordinary news. Posted on the website of the International Congress of Mathematicians that was to be held the following year in Berlin was an announcement that the Holy Grail of mathematics had finally been claimed. The Riemann Hypothesis had been proved. It was news that would have a profound effect. The Riemann Hypothesis is a problem which is central to the whole of mathematics. As they read their email, mathematicians were thrilling to the prospect of understanding one of the greatest mathematical mysteries.

The announcement came in a letter from Professor Enrico Bombieri. One could not have asked for a better, more respected source. Bombieri is one of the guardians of the Riemann Hypothesis and is based at the prestigious Institute for Advanced Study in Princeton, once home to Einstein and Gödel. He is very softly spoken, but mathematicians always listen carefully to anything he has to say.

Bombieri grew up in Italy, where his prosperous family’s vineyards gave him a taste for the good things in life. He is fondly referred to by colleagues as ‘the Mathematical Aristocrat’. In his youth he always cut a dashing figure at conferences in Europe, often arriving in a fancy sports car. Indeed, he was quite happy to fuel a rumour that he’d once come sixth in a twenty-four-hour rally in Italy. His successes on the mathematical circuit were more concrete and led to an invitation in the 1970s to go to Princeton, where he has remained ever since. He has replaced his enthusiasm for rallying with a passion for painting, especially portraits.

But it is the creative art of mathematics, and in particular the challenge of the Riemann Hypothesis, that gives Bombieri the greatest buzz. The Riemann Hypothesis had been an obsession for Bombieri ever since he first read about it at the precocious age of fifteen. He had always been fascinated by properties of numbers as he browsed through the mathematics books his father, an economist, had collected in his extensive library. The Riemann Hypothesis, he discovered, was regarded as the deepest and most fundamental problem in number theory. His passion for the problem was further fuelled when his father offered to buy him a Ferrari if he ever solved it – a desperate attempt on his father’s part to stop Enrico driving his own model.

According to his email, Bombieri had been beaten to his prize. ‘There are fantastic developments to Alain Connes’s lecture at IAS last Wednesday,’ Bombieri began. Several years previously, the mathematical world had been set alight by the news that Alain Connes had turned his attention to trying to crack the Riemann Hypothesis. Connes is one of the revolutionaries of the subject, a benign Robespierre of mathematics to Bombieri’s Louis XVI. He is an extraordinarily charismatic figure whose fiery style is far from the image of the staid, awkward mathematician. He has the drive of a fanatic convinced of his world-view, and his lectures are mesmerising. Amongst his followers he has almost cult status. They will happily join him on the mathematical barricades to defend their hero against any counter-offensive mounted from the ancien régime’s entrenched positions.

Connes is based at France’s answer to the Institute in Princeton, the Institut des Hautes Études Scientifiques in Paris. Since his arrival there in 1979, he has created a completely new language for understanding geometry. He is not afraid to take the subject to the extremes of abstraction. Even the majority of the mathematical ranks who are usually at home with their subject’s highly conceptual approach to the world have balked at the abstract revolution Connes is proposing. Yet, as he has demonstrated to those who doubt the necessity for such stark theory, his new language for geometry holds many clues to the real world of quantum physics. If it has instilled terror in the hearts of the mathematical masses, then so be it.

Connes’s audacious belief that his new geometry could unmask not only the world of quantum physics but explain the Riemann Hypothesis – the greatest mystery about numbers – was met with surprise and even shock. It reflected his disregard for conventional boundaries that he dare venture into the heart of number theory and confront head-on the most difficult outstanding problem in mathematics. Since his arrival on the scene in the mid-nineties, there had been an expectancy in the air that if anyone had the resources to conquer this notoriously difficult problem, it was Alain Connes.

But it was not Connes who appeared to have found the last piece in the complex jigsaw. Bombieri went on to explain that a young physicist in the audience had seen ‘in a flash’ how to use his bizarre world of ‘super-symmetric fermionic—bosonic systems’ to attack the Riemann Hypothesis. Not many mathematicians knew quite what this cocktail of buzzwords meant, but Bombieri explained that it described ‘the physics corresponding to a near-absolute zero ensemble of a mixture of anyons and morons with opposite spins’. It still sounded rather obscure, but then this was after all the solution to the most difficult problem in the history of mathematics, so no one was expecting an easy solution. According to Bombieri, after six days of uninterrupted work and with the help of a new computer language called MISPAR, the young physicist had finally cracked mathematics’ toughest problem.

Bombieri concluded his email with the words, ‘Wow! Please give this the highest diffusion.’ Although it was extraordinary that a young physicist had ended up proving the Riemann Hypothesis, it came as no great surprise. Much of mathematics had found itself entangled with physics over the past few decades. Despite being a problem with its heart in the theory of numbers, the Riemann Hypothesis had for some years been showing unexpected resonances with problems in particle physics.

Mathematicians were changing their travel plans to fly in to Princeton to share the moment. Memories were still fresh with the excitement of a few years earlier when an English mathematician, Andrew Wiles, had announced a proof of Fermat’s Last Theorem in a lecture delivered in Cambridge in June 1993. Wiles had proved that Fermat had been right in his claim that the equation xn + yn = zn has no solutions when n is bigger than 2. As Wiles laid down his chalk at the end of the lecture, the champagne bottles started popping and the cameras began flashing.

Mathematicians knew, however, that proving the Riemann Hypothesis would be of far greater significance for the future of mathematics than knowing that Fermat’s equation has no solutions. As Bombieri had discovered at the tender age of fifteen, the Riemann Hypothesis seeks to understand the most fundamental objects in mathematics – prime numbers.

Prime numbers are the very atoms of arithmetic. The primes are those indivisible numbers that cannot be written as two smaller numbers multiplied together. The numbers 13 and 17 are prime, whilst 15 is not since it can be written as 3 times 5. The primes are the jewels studded throughout the vast expanse of the infinite universe of numbers that mathematicians have explored down the centuries. For mathematicians they instil a sense of wonder: 2, 3, 5, 7, 11, 13, 17, 19, 23, … – timeless numbers that exist in some world independent of our physical reality. They are Nature’s gift to the mathematician.

Their importance to mathematics comes from their power to build all other numbers. Every number that is not a prime can be constructed by multiplying together these prime building blocks. Every molecule in the physical world can be built out of atoms in the periodic table of chemical elements. A list of the primes is the mathematician’s own periodic table. The prime numbers 2, 3 and 5 are the hydrogen, helium and lithium in the mathematician’s laboratory. Mastering these building blocks offers the mathematician the hope of discovering new ways of charting a course through the vast complexities of the mathematical world.

Yet despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge. Look through a list of prime numbers, and you’ll find that it’s impossible to predict when the next prime will appear. The list seems chaotic, random, and offers no clues as to how to determine the next number. The list of primes is the heartbeat of mathematics, but it is a pulse wired by a powerful caffeine cocktail:

The prime numbers up to 100 – mathematics’ irregular heartbeat.

Can you find a formula that generates the numbers in this list, some magic rule that will tell you what the 100th prime number is? This question has been plaguing mathematical minds down the ages. Despite over two thousand years of endeavour, prime numbers seem to defy attempts to fit them into a straightforward pattern. Generations have sat listening to the rhythm of the prime-number drum as it beats out its sequence of numbers: two beats, followed by three beats, five, seven, eleven. As the beat goes on, it becomes easy to believe that random white noise, without any inner logic, is responsible. At the centre of mathematics, the pursuit of order, mathematicians could only hear the sound of chaos.

Mathematicians can’t bear to admit that there might not be an explanation for the way Nature has picked the primes. If there were no structure to mathematics, no beautiful simplicity, it would not be worth studying. Listening to white noise has never caught on as an enjoyable pastime. As the French mathematician Henri Poincaré wrote, ‘The scientist does not study Nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If Nature were not beautiful, it would not be worth knowing, and if Nature were not worth knowing, life would not be worth living.’

One might hope that the prime-number heartbeat settles down after a jumpy start. Not so – things just seem to get worse the higher you count. Here are the primes amongst the 100 numbers either side of 10,000,000. First, those below 10,000,000:

9,999,901 9,999,907, 9,999,929, 9,999,931, 9,999,937, 9,999,943, 9,999,971, 9,999,973, 9,999,991

But look now at how few there are in the 100 numbers above 10,000,000:

10,000,019, 10,000,079.

It is hard to guess at a formula that could generate this kind of pattern. In fact, this procession of primes resembles a random succession of numbers much more than it does a nice orderly pattern. Just as knowing the first 99 tosses of a coin won’t help you much in guessing the result of the 100th toss, so do the primes seem to defy prediction.

Prime numbers present mathematicians with one of the strangest tensions in their subject. On the one hand a number is either prime or it isn’t. No flip of a coin will suddenly make a number divisible by some smaller number. Yet there is no denying that the list of primes looks like a randomly chosen sequence of numbers. Physicists have grown used to the idea that a quantum die decides the fate of the universe, randomly choosing at each throw where scientists will find matter. But it is something of an embarrassment to have to admit that these fundamental numbers on which mathematics is based appear to have been laid out by Nature flipping a coin, deciding at each toss the fate of each number. Randomness and chaos are anathema to the mathematician.

Despite their randomness, prime numbers – more than any other part of our mathematical heritage – have a timeless, universal character. Prime numbers would be there regardless of whether we had evolved sufficiently to recognise them. As the Cambridge mathematician G.H. Hardy said in his famous book A Mathematician’s Apology, ‘317 is a prime not because we think so, or because our minds are shaped in one way or another, but because it is so, because mathematical reality is built that way.’

Some philosophers might take issue with such a Platonist view of the world – this belief in an absolute and eternal reality beyond human existence – but to my mind that is what makes them philosophers and not mathematicians. There is a fascinating dialogue between Alain Connes, the mathematician who featured in Bombieri’s email, and the neurobiologist Jean-Pierre Changeux in Conversations on Mind, Matter and Mathematics. The tension in this book is palpable as the mathematician argues for the existence of mathematics outside the mind, and the neurologist is determined to refute any such idea: ‘Why wouldn’t we see “π = 3.1416” written in gold letters in the sky or “6.02 × 10²³” appear in the reflections of a crystal ball?’ Changeux declares his frustration at Connes’s insistence that ‘there exists, independently of the human mind, a raw and immutable mathematical reality’ and at the heart of that world we find the unchanging list of primes. Mathematics, Connes declares, ‘is unquestionably the only universal language’. One can imagine a different chemistry or biology on the other side of the universe, but prime numbers will remain prime whichever galaxy you are counting in.

In Carl Sagan’s classic novel Contact, aliens use prime numbers to contact life on earth. Ellie Arroway, the book’s heroine, has been working at SETI, the Search for Extraterrestrial Intelligence, listening to the crackle of the cosmos. One night, as the radio telescopes are turned towards Vega, they suddenly pick up strange pulses through the background noise. It takes Ellie no time to recognise the drumbeat in this radio signal. Two pulses are followed by a pause, then three pulses, five, seven, eleven, and so on through all the prime numbers up to 907. Then it starts all over again.

This cosmic drum was playing a music that earthlings couldn’t fail to recognise. Ellie is convinced that only intelligent life could generate this beat: ‘It’s hard to imagine some radiating plasma sending out a regular set of mathematical signals like this. The prime numbers are there to attract our attention.’ Had the alien culture transmitted the previous ten years of alien winning lottery numbers, Ellie couldn’t have distinguished them from the background noise. Even though the list of primes looks as random a list as the lottery winnings, its universal constancy has determined the choice of each number in this alien broadcast. It is this structure that Ellie recognises as the sign of intelligent life.

Communicating using prime numbers is not just science fiction. Oliver Sacks in his book The Man Who Mistook His Wife for a Hat documents twenty-six-year-old twin brothers, John and Michael, whose deepest form of communication was to swap six-digit prime numbers. Sacks tells of when he first discovered them secretly exchanging numbers in the corner of a room: ‘they looked, at first, like two connoisseurs wine-tasting, sharing rare tastes, rare appreciations’. At first, Sacks can’t figure out what the twins are up to. But as soon as he cracks their code, he memorises some eight-digit primes which he drops surreptitiously into the conversation at their next meeting. The twins’ surprise is followed by deep concentration which turns to jubilation as they recognise another prime number. Whilst Sacks had resorted to prime number tables to find his primes, how the twins were generating their primes is a tantalising puzzle. Could it be that these autistic-savants were in possession of some secret formula that generations of mathematicians had missed?

The story of the twins is a favourite of Bombieri’s.

It is hard for me to hear this story without feeling awe and astonishment at the workings of the brain. But I wonder: Do my non-mathematical friends have the same response? Do they have any inkling how bizarre, how prodigious and even other-worldly was the singular talent the twins so naturally enjoyed? Are they aware that mathematicians have been struggling for centuries to come up with a way to do what John and Michael did spontaneously: to generate and recognize prime numbers?

Before anyone could find out how they were doing it, the twins were separated at the age of thirty-seven by their doctors, who believed that their private numerological language had been hindering their development. Had they listened to the arcane conversations that can be heard in the common rooms of university maths departments, these doctors would probably have recommended closing them down too.

It’s likely that the twins were using a trick based on what’s called Fermat’s Little Theorem to test whether a number is prime. The test is similar to the way in which autistic-savants can quickly identify that April 13, 1922, for instance, was a Thursday – a feat the twins performed regularly on TV chat shows. Both tricks depend on doing something called clock or modular arithmetic. Even if they lacked a magic formula for the primes, their skill was still extraordinary. Before they were separated they had reached twenty-digit numbers, well beyond the upper limit of Sacks’s prime number tables.

Like Sagan’s heroine listening to the cosmic prime number beat and Sacks eavesdropping on the prime number twins, mathematicians for centuries had been straining to hear some order in this noise. Like Western ears listening to the music of the East, nothing seemed to make sense. Then, in the middle of the nineteenth century, came a major breakthrough. Bernhard Riemann began to look at the problem in a completely new way. From his new perspective, he began to understand something of the pattern responsible for the chaos of the primes. Underlying the outward noise of the primes was a subtle and unexpected harmony. Despite this great step forward, this new music kept many of its secrets out of earshot. Riemann, the Wagner of the mathematical world, was undaunted. He made a bold prediction about the mysterious music that he had discovered. This prediction is what has become known as the Riemann Hypothesis. Whoever proves that Riemann’s intuition about the nature of this music was right will have explained why the primes give such a convincing impression of randomness.

Riemann’s insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice’s world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann’s glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann’s mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there.

Bombieri’s email of April 7, 1997, promised the beginning of a new era. Riemann’s vision had not been a mirage. The Mathematical Aristocrat had offered mathematicians the tantalising possibility of an explanation for the apparent chaos in the primes. Mathematicians were keen to loot the many other treasures they knew should be unearthed by the solution to this great problem.

A solution of the Riemann Hypothesis will have huge implications for many other mathematical problems. Prime numbers are so fundamental to the working mathematician that any breakthrough in understanding their nature will have a massive impact. The Riemann Hypothesis seems unavoidable as a problem. As mathematicians navigate their way across the mathematical terrain, it is as though all paths will necessarily lead at some point to the same awesome vista of the Riemann Hypothesis.

Many people have compared the Riemann Hypothesis to climbing Mount Everest. The longer it remains unclimbed, the more we want to conquer it. And the mathematician who finally scales Mount Riemann will certainly be remembered longer than Edmund Hillary. The conquest of Everest is marvelled at not because the top is a particularly exciting place to be, but because of the challenge it poses. In this respect the Riemann Hypothesis differs significantly from the ascent of the world’s tallest peak. Riemann’s peak is a place we all want to sit upon because we already know the vistas that will open up to us should we make it to the top. The person who proves the Riemann Hypothesis will have made it possible to fill in the missing gaps in thousands of theorems that rely on it being true. Many mathematicians have simply had to assume the truth of the Hypothesis in reaching their own goals.

The dependence of so many results on Riemann’s challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word ‘hypothesis’ has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. ‘Conjecture’, in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann’s riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated.

By appealing to the Riemann Hypothesis, mathematicians are staking their reputations on the hope that one day someone will prove that Riemann’s intuition was correct. Some go further than just adopting it as a working hypothesis. Bombieri regards it as an article of faith that the primes behave as Riemann’s Hypothesis predicts. It has become virtually a cornerstone in the pursuit of mathematical truth. If, however, the Riemann Hypothesis turns out to be false, it will completely destroy the faith we have in our intuition to sniff out the way things work. So convinced have we become that Riemann was right that the alternative will require a radical revision of our view of the mathematical world. In particular, all the results that we believe exist beyond Riemann’s peak would disappear in a puff of smoke.

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.

The search for patterns

The mathematician’s quest for primes is captured perfectly by one of the tasks we have all faced at school. Given a list of numbers, find the next number. For example, here are three challenges:

1, 3, 6, 10, 15, …

1, 1, 2, 3, 5, 8, 13, …

1, 2, 3, 5, 7, 11, 15, 22, 30, …

Numerous questions spring to the mathematical mind when faced with such lists. What is the rule behind the creation of each list? Can you predict the next number on the list? Can you find a formula that will produce the 100th number on the list without having to calculate the first 99 numbers?

The first sequence of numbers above consists of what are called the triangular numbers. The tenth number on the list is the number of beans required to build a triangle with ten rows, starting with one bean in the first row and ending with ten beans in the last row. So the Nth triangular number is got by simply adding the first N numbers: 1 + 2 + 3 + … + N. If you want to find the 100th triangular number, there is a long laborious method in which you attack the problem head on and add up the first 100 numbers.

Indeed, Gauss’s schoolteacher liked to set this problem for his class, knowing that it always took his students so long that he could take forty winks. As each student finished the task they were expected to come and place their slate tablets with their answer written on it in a pile in front of the teacher. While the other students began labouring away, within seconds the ten-year-old Gauss had laid his tablet on the table. Furious, the teacher thought that the young Gauss was being cheeky. But when he looked at Gauss’s slate, there was the answer – 5,050 – with no steps in the calculation. The teacher thought that Gauss must have cheated somehow, but the pupil explained that all you needed to do was put N = 100 into the formula and you will get the 100th number in the list without having to calculate any other numbers on the list on the way.

Rather than tackling the problem head on, Gauss had thought laterally. He argued that the best way to discover how many beans there were in a triangle with 100 rows was to take a second similar triangle of beans which could be placed upside down on top of the first triangle. Now Gauss had a rectangle with 101 rows each containing 100 beans. Calculating the total number of beans in this rectangle built from the two triangles was easy: there are in total 101 × 100 = 10,100 beans. So one triangle must contain half this number, namely . There is nothing special here about 100. Replace it by N and you get the formula .

The picture overleaf illustrates the argument for the triangle with 10 rows instead of 100.

Instead of directly attacking his teacher’s problem, Gauss had found a different angle from which to view the calculation. Lateral thinking, turning the problem upside down or inside out to see it from a new perspective, is an immensely important theme in mathematical discovery and is one reason why people who can think like the young Gauss make good mathematicians.

The second challenge sequence, 1, 1, 2, 3, 5, 8, 13, …, consists of the so-called Fibonacci numbers. The rule behind this sequence is that each new number is calculated by adding the two previous ones, for example, 13 = 5 + 8. Fibonacci, a mathematician in the thirteenth-century court in Pisa, had struck upon the sequence in relation to the mating habits of rabbits. He had tried to bring European mathematics out of the Dark Ages by proselytising the discoveries of Arabic mathematicians. He failed. Instead, it was the rabbits that immortalised him in the mathematical world. His model of the propagation of rabbits predicted that each new season would see the number of pairs of rabbits grow in a certain pattern. This pattern was based on two rules: each mature pair of rabbits will produce a new pair of rabbits each season, and each new pair will take one season to reach sexual maturity.

An illustration of Gauss’s proof of his formula for the triangular numbers.

But it is not only in the rabbit world that these numbers prevail. This sequence of numbers crops up in all manner of natural ways. The number of petals on a flower invariably is a Fibonacci number, as is the number of spirals in a fir cone. The growth of a seashell over time reflects the progression of the Fibonacci numbers.

Is there a fast formula like Gauss’s formula for the triangular numbers that will produce the 100th Fibonacci number? Again, at first sight it looks as though we might have to calculate all the previous 99 numbers since the way to get the 100th number is to add together the two previous ones. Is it possible that there is a formula that could give us this 100th number just by plugging the number 100 into the formula? This turns out be much trickier, despite the simplicity of the rule for generating these numbers.

The formula for generating the Fibonacci numbers is based upon a special number called the golden ratio, a number which begins 1.618 03… Like the number π, the golden ratio is a number whose decimal expansion continues without end, demonstrating no patterns. Yet it encapsulates what many people down the centuries have regarded as perfect proportions. If you examine the canvases in the Louvre or the Tate Gallery, you’ll find that very often the artist will have chosen a rectangle whose sides are in a ratio of 1 to 1.618 03 … Experiment reveals that a person’s height when compared to the distance from their feet to their belly button favours the same ratio. The golden ratio is a number which appears in Nature in an uncanny fashion. Despite its chaotic decimal expansion, this number also holds the key to generating the Fibonacci numbers. The Nth Fibonacci number can be expressed by a formula built from the Nth power of the golden ratio.

I will leave the third sequence of numbers, 1, 2, 3, 5, 7, 11, 15, 22, 30, …, as a teasing challenge which I will return to later. Its properties helped cement the fame of one of the most intriguing mathematicians of the twentieth century, Srinivasa Ramanujan, who had an extraordinary ability to discover new patterns and formulas in areas of mathematics where others had tried and failed.

It is not just Fibonacci numbers that one finds in Nature. The animal kingdom also knows about prime numbers. There are two species of cicada called Magicicada septendecim and Magicicada tredecim which often live in the same environment. They have a life cycle of exactly 17 and 13 years, respectively. For all but their last year they remain in the ground feeding on the sap of tree roots. Then, in their last year, they metamorphose from nymphs into fully formed adults and emerge en masse from the ground. It is an extraordinary event as, every 17 years, Magicicada septendecim takes over the forest in a single night. They sing loudly, mate, eat, lay eggs, then die six weeks later. The forest goes quiet for another 17 years. But why has each species chosen a prime number of years as the length of their life cycle?

There are several possible explanations. Since both species have evolved prime number life cycles, they will be synchronised to emerge in the same year very rarely. In fact they will have to share the forest only every 221 = 17 × 13 years. Imagine if they had chosen cycles which weren’t prime, for example 18 and 12. Over the same period they would have been in synch 6 times, namely in years 36, 72, 108, 144, 180 and 216. These are the years which share the prime building blocks of both 18 and 12. The prime numbers 13 and 17, on the other hand, allow the two species of cicada to avoid too much competition.

Another explanation is that a fungus developed which emerged simultaneously with the cicadas. The fungus was deadly for the cicadas, so they evolved a life cycle which would avoid the fungus. By changing to a prime number cycle of 17 or 13 years, the cicadas ensured that they emerged in the same years as the fungus less frequently than if they had a non-prime life cycle. For the cicadas, the primes weren’t just some abstract curiosity but the key to their survival.

Evolution might be uncovering primes for the cicadas, but mathematicians wanted a more systematic way to find these numbers. Of all the number challenges it was the list of primes above all others for which mathematicians sought some secret formula. One has to be careful, though, about expecting patterns and order to be everywhere in the mathematical world. Many people throughout history have got lost in the vain attempt to find structure hidden in the decimal expansion of π, one of the most important numbers in mathematics. But its importance has fuelled desperate attempts to discover messages buried in its chaotic decimal expansion. Whilst alien life had used the primes to catch Ellie Arroway’s attention at the beginning of Carl Sagan’s book Contact, the ultimate message of the book is buried deep in the expansion of π, in which a series of O’s and l’s suddenly appears, mapping out a pattern that is meant to reveal ‘there is an intelligence that antedates the universe’. Darren Aronofsky’s film ‘π’ also plays on this popular cultural image.

As a warning to those captivated by the idea of uncovering hidden messages in numbers such as π, mathematicians have been able to prove that most decimal numbers have hidden somewhere in their infinite expansions any sequence of numbers you might be looking for. So there is a good chance that π will contain the computer code for the book of Genesis if you search for long enough. One has to find the right viewpoint from which to look for patterns. π is an important number not because its decimal expansion contains hidden messages. Its importance becomes apparent when it is examined from a different perspective. The same was true of the primes. Armed with his table of primes and his knack for lateral thinking, Gauss was on the lookout for the right angle and viewpoint from which to stare at the primes so that some previously hidden order might emerge from behind the façade of chaos.

Proof, the mathematician’s travelogue

Although finding patterns and structure in the mathematical world is one part of what a mathematician does, the other part is proving that a pattern will persist. The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry. The ancient Greeks were the first to understand that it was possible to prove that certain facts would remain true however far you counted, however many instances you examined.

The mathematical creative process starts with a guess. Often, the guess emerges from the intuition that the mathematician develops after years of exploring the mathematical world, cultivating a feel for its many twists and turns. Sometimes simple numerical experiments reveal a pattern which one might guess will persist for ever. Mathematicians during the seventeenth century, for example, discovered what they believed might be a fail-safe method to test if a number N was prime: calculate 2 to the power N and divide by N – if the remainder is 2 then the number N is a prime. In terms of Gauss’s clock calculator, these mathematicians were trying to calculate 2N on a clock with N hours. The challenge then is to prove whether this guess is right or wrong. It is these mathematical guesses or predictions that the mathematician calls a ‘conjecture’ or ‘hypothesis’.

A mathematical guess only earns the name of ‘theorem’ once a proof has been provided. It is this movement from ‘conjecture’ or ‘hypothesis’ to ‘theorem’ that marks the mathematical maturity of a subject. Fermat left mathematics with a whole slew of predictions. Subsequent generations of mathematicians have made their mark by proving Fermat right or wrong. Admittedly, Fermat’s Last Theorem was always called a theorem and never a conjecture. But that is unusual, and probably came about because Fermat claimed in notes that he scribbled in his copy of Diophantus’s Arithmetica that he had a marvellous proof that was unfortunately too large to write in the margin of the page. Fermat never recorded his supposed proof anywhere, and his marginal comments became the biggest mathematical tease in the history of the subject. Until Andrew Wiles provided an argument, a proof of why Fermat’s equations really had no interesting solutions, it actually remained a hypothesis – merely wishful thinking.

Gauss’s schoolroom episode encapsulates the movement from guess via proof to theorem. Gauss had produced a formula which he predicted would produce any number you wanted on the list of triangular numbers. How could he guarantee that it would work every time? He certainly couldn’t test every number on the list to see whether his formula gave the correct answer, since the list is infinitely long. Instead, he resorted to the powerful weapon of mathematical proof. His method of combining two triangles to make a rectangle guaranteed, without the need for an infinite number of calculations, that the formula would always work. In contrast, the seventeenth-century prime number test based on 2N was finally thrown out of the mathematical court in 1819. The test works correctly for all numbers up to 340, but then declares that 341 is prime. This is where the test fails, since 341 = 11 × 31. This exception wasn’t discovered until Gauss’s clock calculator with 341 hours on the clock face could be used to simplify the analysis of a number like 2³⁴¹, which on a conventional calculator stretches to over a hundred digits.

The Cambridge mathematician G. H. Hardy, author of A Mathematician’s Apology, used to describe the process of mathematical discovery and proof in terms of mapping out distant landscapes: ‘I have always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations.’ Once the mathematician has observed a distant mountain, the second task is then to describe to people how to get there.

You begin in a place where the landscape is familiar and there are no surprises. Within the boundaries of this familiar land are the axioms of mathematics, the self-evident truths about numbers, together with those propositions that have already been proved. A proof is like a pathway from this home territory leading across the mathematical landscape to distant peaks. Progress is bound by the rules of deduction, like the legitimate moves of a chess piece, prescribing the steps you are permitted to take through this world. At times you arrive at what looks like an impasse, and need to take that characteristic lateral step, moving sideways or even backwards to find a way around. Sometimes you need to wait for new tools, like Gauss’s clock calculators, to be invented, so that you can continue your ascent.

In Hardy’s words, the mathematical observer

sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognise it himself. When his pupil also sees it, the research, the argument, the proof is finished.

The proof is the story of the trek and the map charting the coordinates of that journey – the mathematician’s log. Readers of the proof will experience the same dawning realisation as its author. Not only do they finally see the way to the peak, but also they understand that no new development will undermine the new route. Very often a proof will not seek to dot every i and cross every t. It is a description of the journey and not necessarily the re-enactment of every step. The arguments that mathematicians provide as proofs are designed to create a rush in the mind of the reader. Hardy used to describe the arguments we give as ‘gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils’.

The mathematician is obsessed with proof, and will not be satisfied simply with experimental evidence for a mathematical guess. This attitude is often marvelled at and even ridiculed in other scientific disciplines. Goldbach’s Conjecture has been checked for all numbers up to 400,000,000,000,000 but has not been accepted as a theorem. Most other scientific disciplines would be happy to accept this overwhelming numerical data as a convincing enough argument, and move on to other things. If, at a later date, new evidence were to crop up which required a reassessment of the mathematical canon, then fine. If it is good enough for the other sciences, why is mathematics any different?

Most mathematicians would quiver at the thought of such heresy. As the French mathematician André Weil expressed it, ‘Rigour is to the mathematician what morality is to men.’ Part of the reason is that evidence is often quite hard to assess in mathematics. More than any other part of mathematics, the primes take a long time to reveal their true colours. Even Gauss was taken in by overwhelming data in support of a hunch he had about prime numbers, but theoretical analysis later revealed that he had been duped. This is why a proof is essential: first appearances can be deceptive. While the ethos of every other science is that experimental evidence is all that you can truly rely on, mathematicians have learnt never to trust numerical data without proof.

In some respects, the ethereal nature of mathematics as a subject of the mind makes the mathematician more reliant on providing proof to lend some feeling of reality to this world. Chemists can happily investigate the structure of a solid buckminsterfullerene molecule; sequencing the genome presents the geneticist with a concrete challenge; even the physicists can sense the reality of the tiniest subatomic particle or a distant black hole. But the mathematician is faced with trying to understand objects with no obvious physical reality such as shapes in eight dimensions, or prime numbers so large they exceed the number of atoms in the physical universe. Given a palette of such abstract concepts the mind can play strange tricks, and without proof there is a danger of creating a house of cards. In the other scientific disciplines, physical observation and experiment provide some reassurance of the reality of a subject. While other scientists can use their eyes to see this physical reality, mathematicians rely on mathematical proof, like a sixth sense, to negotiate their invisible subject.

Searching for proofs of patterns that have already been spotted is also a great catalyst for further mathematical discovery. Many mathematicians feel that it may be better if these defining problems never get solved because of the wonderful new mathematics encountered along the way. The problems allow for exploration of a kind which forces mathematical pioneers to pass through lands they could never have envisaged at the outset of their journey.

But perhaps the most convincing argument for why the culture of mathematics places such stock in proving that a statement is true is that, unlike the other sciences, there is the luxury of being able to do so. In how many other disciplines is there anything that parallels the statement that Gauss’s formula for triangular numbers will never fail to give the right answer? Mathematics may be an ethereal subject confined to the mind, but its lack of tangible reality is more than compensated for by the certitude that proof provides.

Unlike the other sciences, in which models of the world can crumble between one generation and the next, proof in mathematics allows us to establish with 100 per cent certainty that facts about prime numbers will not change in the light of future discoveries. Mathematics is a pyramid where each generation builds on the achievements of the last without fear of any collapse. This durability is what is so addictive about being a mathematician. For no science other than mathematics can we say that what the ancient Greeks established in their subject holds true today. We may scoff now at the Greeks’ belief that matter was made from fire, air, water and earth. Will future generations look back on the list of 109 atoms that make up Mendeleev’s Periodic Table of elements with as much disdain as we view the Greek model of the chemical world? In contrast, all mathematicians begin their mathematical education by learning what the ancient Greeks proved about prime numbers.

The certainty that proof gives to the mathematician is something that is envied by members of other university departments as much as it is jeered at. The permanence created by mathematical proof leads to the genuine immortality to which Hardy referred. This is often why people surrounded by a world of uncertainty are drawn to the subject. Time after time has the mathematical world offered a refuge for young minds yearning to escape from a real world they cannot cope with.

Our faith in the durability of a proof is reflected in the rules governing the award of Clay’s Millennium Prizes. The prize money is released two years after publication of the proof and with the general acceptance of the mathematical community. Of course, this is no guarantee that there isn’t a subtle error, but it does recognise that we generally believe that errors can be spotted in proofs without waiting many years for new evidence. If there is an error, it must be there on the page in front of us.

Are mathematicians arrogant in believing that they have access to absolute proof? Can one argue that a proof that all numbers are built from primes is as likely to be overthrown as the theory of Newtonian physics or the theory of an indivisible atom? Most mathematicians believe that the axioms that are taken as self-evident truths about numbers will never crumble under future scrutiny. The laws of logic used to build upon these foundations, if applied correctly, will in their view produce proofs of statements about numbers that will never be overturned by new insights. Maybe this is philosophically naive, but it is certainly the central tenet of the sect of mathematics.

There is also the emotional buzz the mathematician experiences in charting new pathways across the mathematical landscape. There is an amazing feeling of exhilaration at discovering a way to reach the summit of some distant peak which has been visible for generations. It is like creating a wonderful story or a piece of music which truly transports the mind from the familiar to the unknown. It is great to make that first sighting of the possible existence of a far-off mountain like Fermat’s Last Theorem or the Riemann Hypothesis. But it doesn’t compare to the satisfaction of navigating the land in between. Even those who follow in the trail of that first pioneer will experience something of the sense of spiritual elevation that accompanied the first moment of epiphany at discovering a new proof. And this is why mathematicians continue to value the pursuit of proof even if they are utterly convinced that something like the Riemann Hypothesis is true. Because mathematics is as much about travelling as it is about arrival.

Is mathematics an act of creation or an act of discovery? Many mathematicians fluctuate between feeling they are being creative and a sense they are discovering absolute scientific truths. Mathematical ideas can often appear very personal and dependent on the creative mind that conceived them. Yet that is balanced by the belief that its logical character means that every mathematician is living in the same mathematical world that is full of immutable truths. These truths are simply waiting to be unearthed, and no amount of creative thinking will undermine their existence. Hardy encapsulates perfectly this tension between creation and discovery that every mathematician battles with: ‘I believe that mathematical reality lies outside us, that our function is to discover or observe it and that the theorems which we prove and which we describe grandiloquently as our “creations” are simply our notes of our observations.’ But at other times he favoured a more artistic description of the process of doing mathematics: ‘Mathematics is not a contemplative but a creative subject,’ he wrote in A Mathematician’s Apology, a book Graham Greene ranked with Henry James’s notebooks as the best account of what it is like to be a creative artist.

Although the primes, and other aspects of mathematics, transcend cultural barriers, much of mathematics is creative and a product of the human psyche. Proofs, the stories mathematicians tell about their subject, can often be narrated in different ways. It is likely that Wiles’s proof of Fermat’s Last Theorem would be as mysterious to aliens as listening to Wagner’s Ring cycle. Mathematics is a creative art under constraints – like writing poetry or playing the blues. Mathematicians are bound by the logical steps they must take in crafting their proofs. Yet within such constraints there is still a lot of freedom. Indeed, the beauty of creating under constraints is that you get pushed in new directions and find things you might never have expected to discover unaided. The primes are like notes in a scale, and each culture has chosen to play these notes in its own particular way, revealing more about historical and social influences than one might expect. The story of the primes is a social mirror as much as the discovery of timeless truths. The burgeoning love of machines in the seventeenth and eighteenth centuries is reflected in a very practical, experimental approach to the primes; in contrast, Revolutionary Europe created an atmosphere where new abstract and daring ideas were brought to bear on their analysis. The choice of how to narrate the journey through the mathematical world is something which is specific to each individual culture.

Euclid’s fables

The first to start telling these stories were the ancient Greeks. They realised the power of proof to forge permanent pathways to mountains in the mathematical world. Once they were reached, no longer was there the fear that these mountains were some distant mathematical mirage. For example, how can we be really sure that there aren’t some rogue numbers out there which can’t actually be built by multiplying together prime numbers? The Greeks were the first to come up with an argument that would leave no doubt in their minds or in the minds of future generations that no such rogue numbers could ever turn up.

Mathematicians often discover proofs by taking a particular instance of the general theory they are trying to prove, and begin by trying to understand why the theory is true for this example. They hope that the argument or recipe that was successful when applied to the example will work regardless of the particular case they chose to analyse. For instance, to prove that every number is a product of primes, start by considering the particular case of the number 140. Suppose you had checked that every number below 140 is either a prime number or the product of prime numbers multiplied together. What about the number 140 itself? Is it possible that this is a rogue number which is neither prime nor equal to a product of prime numbers? First, you would discover that the number is not prime. How would you do this? By showing it could be written as two smaller numbers multiplied together. For example, 140 is 4 × 35. Now we are ‘in’ because we have already confirmed that 4 and 35, numbers lower than our first candidate rogue, 140, can be written as primes multiplied together: 4 is 2 × 2 and 35 is 5 × 7. Piecing this information together, we see that 140 is in fact the product 2 × 2 × 5 × 7. So 140 is not a rogue after all.

The Greeks understood how they could translate this particular example into a general argument that would apply to all numbers. Curiously, their argument begins by asking us to imagine that there are such rogue numbers – ones that are neither prime nor can be written as prime numbers multiplied together. If there are such rogues, then, as we count through the sequence of all the numbers, we must eventually encounter the first of these rogue numbers. We shall call it N (it is sometimes referred to as the minimal criminal). Since this hypothetical number N isn’t a prime number, we must be able to write it as two smaller numbers, A and B, multiplied together. After all, if that weren’t possible, N would be prime.

Since A and B are smaller than N, our choice of N implies that A and B can be written as products of primes. So if we multiply together all the primes coming from A and all the primes coming from B, then we must get the original number, N. We have now shown that N can be written as prime numbers multiplied together, which contradicts our original choice of N. So our original assumption that there were rogue numbers can’t be tenable. Hence every number must be prime or built by multiplying primes.

When I tried this argument out on friends, they felt as if they had been cheated somewhere along the way. There is something slightly slippery about our opening gambit: assume the things you don’t want to exist do exist, and end up proving they don’t. This strategy of thinking the unthinkable became a powerful tool in the Greeks’ construction of proofs. It relies on the logical fact that a statement has to be either true or false. If we assume the statement is false and we get a contradiction, we can infer that our assumption was wrong and deduce that the statement must have been true after all.

The Greek proof appeals to the lazy side of most mathematicians. Instead of being faced with the impossible task of doing an infinite number of explicit calculations to prove that all numbers can be built from primes, the abstract argument captures the essence of every such computation. It’s like knowing how to climb an infinite ladder without physically having to perform the task.

More than any other Greek mathematician, Euclid is regarded as the father of the art of proof. He was part of the research institute that the Greek leader Ptolemy I established in Alexandria around 300 BC. There, Euclid wrote one of the most influential textbooks in all of recorded history: The Elements. In the first part of this book he set down axioms for geometry describing the relationship between points and lines. These axioms were put forward as self-evident truths about the objects of geometry, so that geometry would then act as a mathematical description of the physical world. He went on to use the rules of deduction to produce five hundred theorems of geometry.

The middle part of Euclid’s Elements deals with the properties of numbers, and it is here that we find what many regard as the first truly brilliant piece of mathematical reasoning. In Proposition 20, Euclid explains a simple but fundamental truth about prime numbers: there are infinitely many of them. He begins with the fact that every number can be built by multiplying prime numbers together. On top of this he constructs his next proof. If these prime numbers are the building blocks of all numbers, is it possible, he asks himself, for there to be only a finite number of these building blocks? The Periodic Table of the chemical elements was constructed by Mendeleev, and in its present form it classifies 109 different atoms from which we can build all matter. Could the same be true for prime numbers? What if a mathematical Mendeleev presented Euclid with a list of 109 primes and challenged Euclid to prove that some primes were missing from the list?

Why, for example, can’t all numbers be built simply by multiplying together different combinations of the primes 2, 3, 5 and 7? Euclid thought about how you might look for numbers that aren’t built from any of these primes. You might say, ‘Well that’s easy – just take the next prime, 11.’ This certainly can’t be built from 2, 3, 5 and 7. But sooner or later this strategy is going to fail precisely because, even today, we have no clue about how to guarantee finding where the next prime will be. And because of this unpredictability, Euclid had to try a different tack in his search for a method that would work, regardless of how long the list of primes was.

Whether it was truly Euclid’s own idea or whether he was simply recording ideas that others had dreamt up in Alexandria, we have no way of knowing. Whichever it was, he was able to show how to build a number that couldn’t be built from any finite list of primes that he might be given. Take the primes 2, 3, 5 and 7. Euclid multiplied them together to get 2 × 3 × 5 × 7 = 210, then – and this is his act of genius – he added 1 to this product to get 211. Euclid had constructed this number, 211, in such a way that none of the primes in the list, 2, 3, 5 and 7, would divide into it exactly. By adding 1 to the product, he could guarantee that dividing by a prime on the list would always leave remainder 1.

Now, Euclid knew that all numbers were built by multiplying together primes. So what about 211? Since it can’t be divided by 2, 3, 5 or 7, there had to be some other primes not on the list that built the number 211. In this particular example, 211 is itself a prime. Euclid was not claiming that the number he built would always be prime – only that it was a number that was built out of primes that were not on the list that our mathematical Mendeleev was offering us.

For example, what if one claimed that all numbers could be built from the finite list of primes 2, 3, 5, 7, 11 and 13. Euclid’s number built from these primes is 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031. This number is not a prime. All Euclid was saying was that, given any list containing finitely many primes, he could produce a number that had to be built out of primes that were not on the list. In this particular case the primes you need are 59 and 509. But in general, Euclid had no way of knowing how to find the precise value of these new primes. He knew only that they must exist.

It was a wonderful argument. Euclid had no idea how to produce primes explicitly, but he could prove why they would never run dry. It is striking that we do not know whether infinitely many of Euclid’s numbers themselves are prime, even though they are sufficient to prove that there must be an infinite number of primes. With Euclid’s proof, gone was the chance of fitting together a Periodic Table listing all the primes or of discovering some prime number genome coding billions of them. No simple butterfly collecting would ever allow us to understand these numbers. Here, then, was the ultimate challenge: the mathematician, armed with a limited weaponry, pitched against the infinite expanse of prime numbers. How could we possibly chart a path through such an infinite chaotic jumble and find some pattern which might predict their behaviour?

Hunting for primes

Generations have striven without success to improve on Euclid’s understanding of the primes, and there have been many intriguing speculations. But as Cambridge don Hardy liked to say, ‘Every fool can ask questions about prime numbers that the wisest man cannot answer.’ The Twin Primes Conjecture, for example, asks whether there are infinitely many primes p such that the number p + 2 is also prime. An example of such a pair is 1,000,037 and 1,000,039. (Note that this is the closest that two primes numbers can be, since N and N + 1 cannot both be prime – except when N = 2 – because at least one of these numbers is divisible by 2.) Might Sacks’s autistic-savant twins have had an extra facility for finding these twin primes? Euclid proved two millennia ago that there are infinitely many primes, but no one knows whether there might be some number beyond which there are no longer such close primes. We believe that there are infinitely many twin primes. Guesses are one thing, but proof remains the ultimate goal.

Mathematicians tried, with varying degrees of success, to come up with formulas that, even if they don’t generate all prime numbers, do produce a list of primes. Fermat thought he had one. He guessed that if you raise 2 to the power 2N and add 1, the resulting number is a prime. This number is called the Nth Fermat number. For example, taking N = 2 and raising 2 to the power 2² = 4, you get 16. Add 1 and you get the prime number 17, the second Fermat number. Fermat thought that his formula would always give him a prime number, but it turned out to be one of the few guesses that he got wrong. The Fermat numbers get very large very quickly. Even the fifth Fermat number has 10 digits, and was out of Fermat’s computational reach. It is the first Fermat number which is not prime, being divisible by 641.

Fermat’s numbers were very dear to Gauss’s heart. The fact that 17 is one of Fermat’s primes is the key to why Gauss could construct his perfect 17-sided shape. In his great treatise Disquisitiones Arithmeticae, Gauss shows why it is that, if the Nth Fermat number is a prime, you can make a geometric construction of an N-sided shape only using a straight edge and compass. The fourth Fermat number, 65,537, is prime, so with these very basic instruments it is possible to construct a perfect 65,537-sided figure.

Fermat’s numbers have failed to throw up more than four primes to date, but he had more success in uncovering some of the very special properties that prime numbers have. Fermat discovered a curious fact about those prime numbers that leave remainder 1 on division by 4 – examples are 5, 13, 17 and 29. Such prime numbers can always be written as the sum of two squares – for example, 29 = 2² + 5². This was another of Fermat’s teases. Although he claimed to have a proof, he failed to record much of the details.

On Christmas Day, 1640, Fermat wrote of his discovery – that certain primes could be expressed as the sum of two squares – in a letter to a French monk called Marin Mersenne. Mersenne’s interests were not confined to liturgical matters. He loved music and was the first to develop a coherent theory of harmonics. He also loved numbers. Mersenne and Fermat corresponded regularly about their mathematical discoveries, and Mersenne broadcast many of Fermat’s claims to a wider audience. Mersenne became renowned for his role as an international scientific clearing house through which mathematicians could disseminate their ideas.

Just as generations had been captivated by the search for order in the primes, Mersenne too had caught the bug. Although he couldn’t see a way to find a formula that would produce all the primes, he did come across a formula that in the long run has proved far more successful at finding primes than Fermat’s formula has. Like Fermat, he started by considering powers of 2. But instead of adding 1, as Fermat had, Mersenne decided to subtract 1 from the answer. So, for example, 2³ − 1 = 8 − 1 = 7, a prime number. Maybe Mersenne’s musical intuition was coming to his aid. Doubling the frequency of a note takes the note up an octave, so powers of 2 produce harmonic notes. You might expect a shift of 1 to sound a very dissonant note, not compatible with any previous frequency – a ‘prime note’.

Mersenne quickly discovered that his formula wasn’t going to yield a prime every time. For example, 2⁴ − 1 = 15. Mersenne realised that if n was not prime then there was no chance that 2n − 1 was going to be prime. But now he boldly claimed that, for n up to 257, 2n − 1 would be prime precisely if n was one of the following numbers: 2, 3, 5, 7, 13, 19, 31, 67, 127, 257. He had discovered that even if n was prime, it still annoyingly didn’t guarantee that his number 2n − 1 would be prime. He could calculate 2¹¹ − 1 by hand and get 2,047, which is 23 × 89. Generations of mathematicians marvelled at Mersenne’s ability to assert that a number as large as 2²⁵⁷ − 1 was prime. This number has 77 digits. Did the monk have access to some mystical arithmetic formula that told him why this number, beyond any human computational abilities, was prime?

Mathematicians believe that if one continues Mersenne’s list, there will be infinitely many choices for n which will make Mersenne’s numbers 2n − 1 into prime numbers. But we are still missing a proof that this guess is true. We are still waiting for a modern day Euclid to prove that Mersenne’s primes never run dry. Or perhaps this far-off peak is just a mathematical mirage.

Many mathematicians of Fermat and Mersenne’s generation had played around with interesting numerological properties of the primes, but their methods did not match up to the ancient Greek ideal of proof. This explains in part why Fermat gave no details of many of the proofs he claimed to have discovered. There was a distinct lack of interest during this period in providing such logical explanations. Mathematicians were quite content with a more experimental approach to their subject, where in an increasingly mechanised world results were justified by their practical applications. In the eighteenth century, however, there arrived a mathematician who would rekindle a sense of the value of proof in mathematics. The Swiss mathematician Leonhard Euler, born in 1707, came up with explanations for many of the patterns that Fermat and Mersenne had discovered but failed to account for. Euler’s methods would later play a significant role in opening new theoretical windows onto our understanding of the primes.

Euler, the mathematical eagle

The mid-eighteenth century was a time of court patronage. This was pre-Revolutionary Europe, when countries were ruled by enlightened despots: Frederick the Great in Berlin, Peter the Great and Catherine the Great in St Petersburg, Louis XV and Louis XVI in Paris. Their patronage supported the academies that drove the intellectual development of the Enlightenment, and indeed they saw it as a mark of their standing that they be surrounded in their courts by intellectuals. And they were well aware of the potential of the sciences and mathematics to boost the military and industrial capabilities of their countries.

Euler was the son of a clergyman who hoped that his son would join him in the church. The young Euler’s precocious mathematical talents, however, had brought him to the notice of the powers that be. Euler was soon being courted by the academies throughout Europe. He had been tempted to join the Academy in Paris, which by this time had become the world’s centre of mathematical activity. He chose instead to accept an offer made to him in 1726 to join the Academy of Sciences in St Petersburg, the capstone for Peter the Great’s campaign to improve education in Russia. He would be joining friends from Basel who had stimulated his interest in mathematics as a child. They wrote to Euler from St Petersburg asking whether he could bring from Switzerland fifteen pounds of coffee, one pound of the best green tea, six bottles of brandy, twelve dozen fine tobacco pipes and a few dozen packs of playing cards. Laden down with gifts, it took the young Euler seven weeks to complete the long journey by boat, on foot and by post wagon, and in May 1727 he finally arrived in St Petersburg to pursue his mathematical dreams. His subsequent output was so extensive that the St Petersburg Academy was still publishing material that had been housed in their archives some fifty years after Euler’s death in 1783.

The role of the court mathematician is perfectly illustrated by a story that was told of Euler’s time in St Petersburg. Catherine the Great was hosting the famous French philosopher and atheist Denis Diderot. Diderot was always rather damning of mathematics, declaring that it added nothing to experience and served only to draw a veil between human beings and nature. Catherine, though, quickly tired of her guest, not because of Diderot’s disparaging views on mathematics but rather his tiresome attempts to rattle the religious faith of her courtiers. Euler was promptly called to court to assist in silencing the insufferable atheist. In appreciation of her patronage, Euler duly consented and addressed Diderot in serious tones before the assembled court: ‘Sir, (a + bn)/n = x, hence God exists; reply.’ Diderot is reported to have retreated in the light of such a mathematical onslaught.

This anecdote, told by the famous English mathematician Augustus De Morgan in 1872, had probably been embroidered for popular consumption and is a reflection more of the fact that most mathematicians enjoy putting down philosophers. But it does show how the royal courts of Europe had not considered themselves complete without a smattering of mathematicians amongst the ranks of astronomers, artists and composers.

Catherine the Great was interested not so much in mathematical proofs of the existence of God, but rather in Euler’s work on hydraulics, ship construction and ballistics. The Swiss mathematician’s interests ranged far and wide over the mathematics of the day. As well as military mathematics, Euler also wrote on the theory of music, but ironically his treatise was regarded as too mathematical for musicians and too musical for mathematicians.

One of his popular triumphs was the solution of the Problem of the Bridges of Königsberg. The River Pregel, known now as the Pregolya, runs through Königsberg, which in Euler’s day was in Prussia (it’s now in Russia, and called Kaliningrad). The river divides, creating two islands in the centre of the town, and the Königsbergers had built seven bridges to span it (see overleaf).

It had become a challenge amongst the citizens to see if anyone could walk around the town, crossing each bridge once and only once, and return to their starting point. It was Euler who eventually proved in 1735 that the task was impossible. His proof is often cited as the beginning of topology, where the actual physical dimensions of the problem are not relevant. It was the network of connections between different parts of the town that was important to Euler’s solution, and not their actual locations or distances apart – the map of the London Underground illustrates this principle.

It was numbers above all that captivated Euler’s heart. As Gauss would write:

The peculiar beauties of these fields have attracted all those who have been active there; but none has expressed this so often as Euler, who, in almost every one of his many papers on number-theory, mentions again and again his delight in such investigations, and the welcome change he finds there from tasks more directly related to practical applications.

The bridges of Konigsberg.

Euler’s passion for number theory had been stimulated by correspondence with Christian Goldbach, an amateur German mathematician who was living in Moscow and unofficially employed as secretary of the Academy of Sciences in St Petersburg. Like the amateur mathematician Mersenne before him, Goldbach was fascinated by playing around with numbers and doing numerical experiments. It was to Euler that Goldbach communicated his conjecture that every even number could be written as a sum of two primes. Euler in return would write to Goldbach to try out many of the proofs he had constructed to confirm Fermat’s mysterious catalogue of discoveries. In contrast to Fermat’s reticence in keeping his supposed proofs a secret from the world, Euler was happy to show off to Goldbach his proof of Fermat’s claim that certain primes can be written as the sum of two squares. Euler even managed to prove an instance of Fermat’s Last Theorem.

Despite his passion for proof, Euler was still very much an experimental mathematician at heart. Many of his arguments flew close to the mathematical wind, containing steps that weren’t completely rigorous. That did not concern him if it led to interesting new discoveries. He was a mathematician of exceptional computational skill and very adept at manipulating mathematical formulas until strange connections emerged. As the French academician François Arago observed, ‘Euler calculated without apparent effort, as men breathe, or eagles sustain themselves in the wind.’

Above all else, Euler loved calculating prime numbers. He produced tables of all the primes up to 100,000 and a few beyond. In 1732, he was also the first to show that Fermat’s formula for primes, , broke down when N = 5. Using new theoretical ideas, he managed to show how to crack this ten-digit number into a product of two smaller numbers. One of his most curious discoveries was a formula that seemed to generate an uncanny number of primes. In 1772, he calculated all the answers that you get when you feed the numbers from 0 to 39 into the formula x² + x + 41. He got the following list:

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1,033, 1,097, 1,163, 1,231, 1,301, 1,373, 1,447, 1,523, 1,601

It seemed bizarre to Euler that you could generate so many primes with this formula. He realised that the process would have to break down at some point. It might already be clear to you that when you input 41, the output has to be divisible by 41. Also, for x = 40 you get a number which is not prime.

Nonetheless, Euler was quite struck by his formula’s ability to produce so many primes. He began to wonder what other numbers might work instead of 41. He discovered that in addition to 41 you could also choose q = 2, 3, 5, 11, 17, and the formula x² + x + q would spit out primes when fed numbers from 0 to q − 2.

But finding such a simple formula for generating all the primes was beyond even the great Euler. As he wrote in 1751, ‘There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.’ It seems paradoxical that the fundamental objects on which we build our order-filled world of mathematics should behave so wildly and unpredictably.

It would turn out that Euler had been sitting on an equation that would break the prime number deadlock. But it would take another hundred years, and another great mind, to show what Euler could not. That mind belonged to Bernhard Riemann. It was Gauss, though, who by initiating another of his classic lateral moves, would eventually inspire Riemann’s new perspective.

Gauss’s guess

If centuries of searching had failed to unearth some magic formula which would generate the list of prime numbers, perhaps it was time to adopt a different strategy. This was what the fifteen-year-old Gauss was thinking in 1792. He had been given a book of logarithms as a present the previous year. Until a few decades ago, tables of logarithms were familiar to every teenager doing calculations in the schoolroom. With the advent of pocket calculators, they lost their place as an essential tool in everyday life, but several hundred years ago every navigator, banker and merchant would have been exploiting these tables to turn difficult multiplication into simple addition. Included at the back of Gauss’s new book was a table of prime numbers. It was uncanny that primes and logarithms should appear together, because Gauss noticed after extensive calculations that there seemed to be a connection between these two seemingly unrelated topics.

The first table of logarithms was conceived in 1614 in an age when sorcery and science were bedfellows. Their creator, the Scottish Baron John Napier, was regarded by local residents as a magician who dealt in the dark arts. He skulked around his castle dressed in black, a jet-black cock perched on his shoulder, muttering that his apocalyptic algebra foretold that the Last Judgement would fall between 1688 and 1700. But as well as applying his mathematical skills to the practice of the occult, he also uncovered the magic of the logarithm function.

If you feed a number, say 100, into your calculator and then press the ‘log’ button, the calculator spits out a second number, the logarithm of 100. What your calculator has done is to solve a little puzzle: it has looked for the number x that makes the equation 10x = 100 correct. In this case the calculator outputs the answer 2. If we input 1,000, a number ten times larger than 100, then the new answer output by your calculator is 3. The logarithm goes up by 1. Here is the essential character of the logarithm: it turns multiplication into addition. Each time we multiply the input by 10, we get the new output by adding 1 to the previous answer.

It was a fairly major step for mathematicians to realise that they could talk about logarithms of numbers which weren’t whole-number powers of 10. For example, Gauss would have been able to look up in his tables the logarithm of 128 and find that raising 10 to the power 2.107 21 would get him pretty close to 128. These calculations are what Napier had collected together in the tables that he had produced in 1614.

Tables of logarithms helped to accelerate the world of commerce and navigation that was blossoming in the seventeenth century. Because of the dialogue that logarithms created between multiplication and addition, the tables helped to convert a complicated problem of multiplying together two large numbers into the simpler task of adding their logarithms. To multiply together large numbers, merchants would add together the logarithms of the numbers and then use the log tables in reverse to find the result of the original multiplication. The increase in speed that the sailor or seller would gain via these tables might save the wrecking of a ship or the collapse of a deal.

But it was the supplementary table of prime numbers at the back of his book of logarithms that fascinated the young Gauss. In contrast to the logarithms, these tables of primes were nothing more than a curiosity to those interested in the practical application of mathematics. (Tables of primes constructed in 1776 by Antonio Felkel were considered so useless that they ended up being used for cartridges in Austria’s war with Turkey!) The logarithms were very predictable; the primes were completely random. There seemed no way to predict when to expect the first prime after 1,000, for example.

The important step Gauss took was to ask a different question. Rather than attempting to predict the precise location of the next prime, he tried instead to see whether he could at least predict how many primes there were in the first 100 numbers, the first 1,000 numbers, and so on. If one took any number N, was there a way to estimate how many primes one would expect to find amongst the numbers from 1 to N? For example, there are twenty-five prime numbers up to 100. So you have a one in four chance of getting a prime if you choose a number at random between 1 and 100. How does this proportion change if we look at the primes from 1 to 1000, or 1 to 1,000,000? Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.

If we look at the table overleaf of values of the number of primes up to various powers of ten, based on more modern calculations, this regularity becomes apparent.

This table, which contains much more information than was available to Gauss, shows us more clearly the regularity that Gauss discovered. It is in the last column that the pattern manifests itself. This column represents the proportion of prime numbers amongst all the numbers being considered. For example, 1 in 4 numbers are prime counting up to 100, so in that interval you will need to count, on average, 4 to get from one prime to the next. Of the numbers up to 10 million, 1 in 15 are prime. (So, for example, there is a 1 in 15 chance that a seven-digit telephone number is a prime.) For N greater than 10,000, this last column seems to be just increasing by about 2.3 each time.

So every time Gauss multiplied by 10, he had to add about 2.3 to the ratio of the primes to all the numbers. This link between multiplication and addition is precisely the relationship embodied in a logarithm. Gauss, with his book of logarithms, would have found this connection staring him in the face.

The reason why the proportion of primes was increasing by 2.3 rather than 1 every time Gauss multiplied by 10 is because primes favour logarithms based not on powers of 10 but on powers of a different number. Pressing the ‘log’ button on your calculator when you input 100 produced the answer 2, which is the solution to the equation 10x = 100. But there is nothing that says we have to have 10 as the number to raise to the power x. It is our obsession with our ten fingers which makes 10 so appealing. The choice of the number 10 is called the base of the logarithm. We can talk about the logarithm of a number to a base other than 10. For example, the logarithm of 128 to base 2 rather than base 10 requires us to solve a different puzzle, to find a number x so that 2x = 128. If we had a ‘log to base 2’ button on our calculator, we could press it and get the answer 7, because we need to raise 2 to the power of 7 to get up to 128: 2⁷ = 128.

What Gauss discovered is that primes can be counted using logarithms to the base of a special number, called e, which to twelve decimal places is 2.718 281 828459 … (like π, it has an infinite decimal expansion with no repeating patterns), e turns out to be as important in mathematics as the number π, and occurs all over the mathematical world. This is why logarithms to the base e are called ‘natural’ logarithms.

The table that Gauss had made at the age of fifteen led him to the following guess. For the numbers from 1 to N roughly 1 out of every log(N) numbers will be prime (where log(N) denotes the logarithm of N to the base e). He could then estimate the number of primes from 1 to N as roughly N/log(N). Gauss was not claiming that this magically gave him an exact formula for the number of primes up to N – it just seemed to provide a very good ballpark estimate.

It was a similar philosophy that he would later apply in his rediscovery of Ceres. His astronomical method made a good prediction for a small region of space to look at, given the data that had been recorded. Gauss had taken the same approach for the primes. Generations had become obsessed with trying to predict the precise location of the next prime, with producing formulas that would generate prime numbers. By not getting hung up on the minutiae of which number was prime or not, Gauss had hit on some sort of pattern. By stepping back and asking the broader question of how many primes there are up to a million rather than precisely which numbers are prime, a strong regularity seemed to emerge.

Gauss had made an important psychological shift in looking at the primes. It was as if previous generations had listened to the music of the primes note by note, unable to hear the whole composition. By concentrating instead on counting how many primes there were as one counted higher, Gauss found a new way to hear the dominant theme.

Following Gauss, it has become customary to denote the number of primes we find in the numbers from 1 to N by the symbol π(N) (this is just a name for this count and has nothing to do with the number π). It is perhaps unfortunate that Gauss used a symbol that makes one think of circles and the number 3.1415 … Think of this instead as a new button on your calculator. You input a number N and press the π(N) button, and the calculator outputs the number of primes up to N. So, for example, π(100) = 25, the number of primes up to 100, and π(1,000) = 168.

Notice that you can still use this new ‘count primes’ button to identify precisely when you get a prime. If you input the number 100 and press this button, counting primes between 1 and 100, you get the answer 25. If you now input the number 101, the answer will go up by 1 to 26 primes, and this means that 101 is a new prime number. So whenever there is a difference between π(N) and π(N +1), you know that N + 1 must be a new prime.

To reveal quite how stunning Gauss’s pattern was, we can look at a graph of the function π(N) counting the number of primes from 1 to N. Here’s a graph of π(N) for numbers N from 1 to 100:

The prime number staircase – the graph counts the cumulative number of primes up to 100.

On this small scale the result is a jumpy staircase, where it is hard to predict how long one has to wait for the next step to appear. We are still seeing at this scale the minutiae of the primes, the individual notes.

Now step back and look at the same function where N is taken over a much wider range of numbers, say counting the primes up to 100,000.

The prime number staircase counting primes up to 100,000.

The individual steps themselves become insignificant and we end up seeing the overarching trend of this function creeping up. This was the big theme that Gauss had heard and could imitate using the logarithm function.

The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, ‘You have no idea how much poetry there is in a table of logarithms.’

Why Gauss was so reticent about something so momentous is a mystery. It is true that he had only found early evidence of some connection between primes and the logarithm function. He knew that he had absolutely no explanation or proof of why these two things should have anything to do with each other. It wasn’t clear that this pattern might not suddenly disappear as you counted higher. Gauss’s reluctance to announce unproved results marked a turning point in mathematical history. Although the Greeks had introduced the idea of proof as an important component of the mathematical process, mathematicians before Gauss’s time were much more interested in scientific speculation about mathematics. If the mathematics worked, they weren’t too concerned about a rigorous justification of why it worked. Mathematics was still the tool of the other sciences.

Gauss broke from the past by stressing the value of proof. For him, presenting proofs was the primary goal of the mathematician, an ethos which has remained fundamental to this day. Without a proof of the connection between logarithms and primes, Gauss’s discovery was worthless to him. The freedom that the patronage of the Duke of Brunswick permitted him meant he could be quite choosy, almost indulgent about the mathematics he produced. His prime motivation was not fame and recognition but a personal understanding of the subject he loved. His seal bore the motto Pauca sed matura (‘Few but ripe’). Unless a result had reached full maturity it remained an entry in his diary or a doodle at the back of his table of logarithms.

For Gauss, mathematics was a private pursuit. He even encrypted entries in his diary using his own secret language. Some of them are easy to unravel. For example, on July 10, 1796, Gauss wrote Archimedes’ famous declaration ‘Eureka!’ followed by the equation

num = Δ + Δ + Δ

which represents his discovery that every number can be written as the sum of three numbers from the list of triangular numbers, 1, 3, 6, 10, 15, 21, 28, …, those numbers for which Gauss had produced his schoolroom formula. For example, 50 = 1 + 21 + 28. But other entries remain a complete mystery. No one has been able to unravel what Gauss meant when he wrote on October 11, 1796, ‘Vicimus GEGAN’. Some have blamed Gauss’s failure to disseminate his discoveries for holding back the development of mathematics by half a century. If he had bothered to explain half of what he had discovered and not been so cryptic in the explanations he did offer, mathematics might have advanced at a quicker pace.

Some people believe that Gauss kept his results to himself because the Paris Academy had rejected his great treatise on number theory, Disquisitiones Arithmeticae, as obscure and dense. Having been stung by rejection, to protect himself from any further humiliation he insisted that every last piece of the mathematical jigsaw be in place before he would consider publishing anything. Disquisitiones Arithmeticae did not receive immediate acclaim partly because Gauss continued to be cryptic even in the work he did expose to public view. He always claimed that mathematics was like a piece of architecture. An architect never leaves the scaffolding for people to see how the building was constructed. This was not a philosophy that helped mathematicians to penetrate Gauss’s mathematics.

But there were other reasons why Paris was not as receptive to Gauss’s ideas as he had hoped. By the end of the eighteenth century, mathematics in Paris was becoming ever more dedicated to serving the demands of an increasingly industrialised state. The Revolution of 1789 had shown Napoleon the need for a more centralised teaching of military engineering, and he responded by founding the École Polytechnique to further his war aims. ‘The advancement and perfection of mathematics are intimately connected with the prosperity of the State,’ Napoleon declared. French mathematics was dedicated to solving problems of ballistics and hydraulics. But despite this emphasis on the practical needs of the state, Paris still boasted some of the leading pure mathematicians in Europe.

One of the great authorities in Paris was Adrien-Marie Legendre, who was born twenty-five years before Gauss. Portraits of Legendre depict a rather puffed-up gentleman with a round, chubby face. In contrast to Gauss, Legendre came from a wealthy family but he had lost his fortune during the Revolution and been forced to rely instead on his mathematical talents for his livelihood. He too was interested in the primes and number theory, and in 1798, six years after Gauss’s childhood calculations, he announced his discovery of the experimental connection between primes and logarithms.

Although it would later be proved that Gauss had indeed beaten Legendre to the discovery, Legendre did nonetheless improve on the estimate for the number of primes up to N. Gauss had guessed that there were roughly N/log(N) primes up to N. Although this was close, it was found to gradually drift away from the true number of primes as N got larger and larger. Here is a comparison of Gauss’s childhood guess, shown as the lower plot, with the true number of primes, the upper plot:

A comparison between Gauss’s guess and the true number of primes.

This graph reveals that although Gauss was on to something, there still seemed to be room for improvement.

Legendre’s improvement was to replace the approximation N/log(N) by

thus introducing a small correction which had the effect of shifting Gauss’s curve up towards the true number of primes. As far as the values of these functions that were then within computational reach, it was impossible to distinguish the two graphs of π(N) and Legendre’s estimate. Legendre, steeped in the prevailing preoccupation with the practical application of mathematics, was much less reluctant to stick his neck out and make some prediction about the connection between primes and logarithms. He was not a man scared to circulate unproven ideas, or even proofs with gaps in them. In 1808 he published his guess at the number of primes in a book about number theory entitled Théorie des Nombres.

The controversy over who first discovered the connection between primes and logarithms led to a bitter dispute between Legendre and Gauss. It was not limited to the argument about primes – Legendre even claimed that he had been the first to discover Gauss’s method for establishing the motion of Ceres. Time and again, Legendre’s assertion that he had uncovered some mathematical truth would be countered by an announcement by Gauss that he had already plundered that particular treasure. Gauss commented in a letter of July 30, 1806, written to an astronomical colleague named Schumacher, ‘It seems to be my fate to concur in nearly all my theoretical works with Legendre.’

In his lifetime, Gauss was too proud to get involved in open battles of priority. When Gauss’s papers and correspondence were examined after his death, it became clear that due credit invariably went to Gauss. It wasn’t until 1849 that the world learnt that Gauss had beaten Legendre to the connection between primes and logarithms, which Gauss disclosed in a letter to a fellow mathematician and astronomer, Johann Encke, written on Christmas Eve of that year.

Given the data available at the start of the nineteenth century, Legendre’s function was much better than Gauss’s formula as an approximation to the number of primes up to some number N. But the appearance of the rather ugly correction term 1.083 66 made mathematicians believe that something better and more natural must exist to capture the behaviour of the prime numbers.

Such ugly numbers may be commonplace in other sciences, but it is remarkable how often the mathematical world favours the most aesthetic possible construction. As we shall see, Riemann’s Hypothesis can be interpreted as an example of a general philosophy among mathematicians that, given a choice between an ugly world and an aesthetic one. Nature always chooses the latter. It is a constant source of amazement for most mathematicians that mathematics should be like this, and explains why they so often get wound up about the beauty of their subject.

It is perhaps not surprising that in later life Gauss further refined his guess and arrived at an even more accurate function, one which was also much more beautiful. In the same Christmas Eve letter that Gauss wrote to Encke, he explains how he subsequently discovered how to go one better than Legendre’s improvement. What Gauss did was to go back to his very first investigations as a child. He had calculated that amongst the first 100 numbers, 1 in 4 are prime. When he considered the first 1,000 numbers the chance that a number is prime went down to 1 in 6. Gauss realised that the higher you count, the smaller the chance that a number will be prime.

So Gauss formed a picture in his mind of how Nature might have decided which numbers were going to be prime and which were not. Since their distribution looked so random, might tossing a coin not be a good model for choosing primes? Did Nature toss a coin – heads it’s prime, tails it’s not? Now, thought Gauss, the coin could be weighted so that instead of landing heads half the time, it lands heads with probability 1/log(N). So the probability that the number 1,000,000 is prime should be interpreted as l/log(1,000,000), which is about 1/15. The chances that each number N is a prime gets smaller as N gets bigger because the probability 1/log(N) of coming up heads is getting smaller.

This is just a heuristic argument because 1,000,000 or any other particular number is either prime or it isn’t. No toss of a coin can alter that. Although Gauss’s mental model was useless at predicting whether a number is prime, he found it very powerful at making predictions about the less specific question of how many primes one might expect to encounter as one counted higher. He used it to estimate the number of primes you should get after tossing the prime number coin N times. With a normal coin which lands heads with probability ½, the number of heads should be ½N. But the probability with the prime number coin is getting smaller with each toss. In Gauss’s model the number of primes is predicted to be

Gauss actually went one step further to produce a function which he called the logarithmic integral, denoted by Li(N). The construction of this new function was based on a slight variation of the above sum of probabilities, and it turned out to be stunningly accurate.

By the time Gauss, in his seventies, wrote to Encke, he had constructed tables of primes up to 3,000,000. ‘I very often used an idle quarter of an hour to count through another chiliad [an interval of 1,000 numbers] here and there’ in his search for prime numbers. His estimate for primes less than 3,000,000 using his logarithmic integral is a mere seven hundredths of 1 per cent off the mark. Legendre had managed to massage his ugly formula to match π(N) for small N, so with the data available at the time it looked as if Legendre’s formula was superior. As more extensive tables began to be drawn up, they revealed that Legendre’s estimate grew far less accurate for primes beyond 10,000,000. A professor at the University of Prague, Jakub Kulik, spent twenty years single-handedly constructing tables of primes for numbers up to 100,000,000. The eight volumes of this gargantuan effort, completed in 1863, were never published but were deposited in the archives of the Academy of Sciences in Vienna. Although the second volume has gone astray, the tables already revealed that Gauss’s method, based on his Li(N) function, was once again outstripping Legendre’s. Modern tables show just how much better Gauss’s intuition was. For example, his estimate for the number of primes up to 10¹⁶ (i.e. 10,000,000,000,000,000) deviates from the correct value by just one ten-millionth of 1 per cent, whilst Legendre’s is now off by one-tenth of 1 per cent. Gauss’s theoretical analysis had triumphed over Legendre’s attempts to manipulate his formula to match the available data.

Gauss noticed a curious feature about his method. Based on what he knew about the primes up to 3,000,000, he could see that his formula Li(N always appeared to overestimate the number of primes. He conjectured that this would always be the case. And who wouldn’t back Gauss’s hunch, now that modern numerical evidence confirms Gauss’s conjecture up to 10¹⁶? Certainly any experiment that produced the same result 10¹⁶ times would be regarded as pretty convincing evidence in most laboratories – but not in the mathematician’s. For once, one of Gauss’s intuitive guesses turned out to be wrong. But although mathematicians have now proved that eventually π(N) must sometimes overtake Li(N), no one has ever seen it happen because we can’t count far enough yet.

A comparison of the graphs of π(N) and Li(N) shows such a good match that over a large range it is barely possible to distinguish the two. I should stress that a magnifying glass applied to any portion of such a picture will show that the functions are different. The graph of π(N) looks like a staircase, whilst Li(N) is a smooth graph with no sharp jumps.

Gauss had uncovered evidence of the coin that Nature had tossed to choose the primes. The coin was weighted so that a number N has a chance of 1 in log(N) of being prime. But he was still missing a method of predicting precisely the tosses of the coin. That would take the insight of a new generation.

By shifting his perspective, Gauss had perceived a pattern in the primes. His guess became known as the Prime Number Conjecture. To claim Gauss’s prize, mathematicians had to prove that the percentage error between Gauss’s logarithmic integral and the real number of primes gets smaller and smaller the further you count. Gauss had seen this far-off mountain peak, but it was left to future generations to provide a proof, to reveal the pathway to the summit, or to unmask the connection as an illusion.

Many blame the appearance of Ceres for distracting Gauss from proving the Prime Number Conjecture himself. The overnight fame he received at the age of twenty-four steered him towards astronomy, and mathematics no longer had pride of place. When his patron, Duke Ferdinand, was killed by Napoleon in 1806, Gauss was forced to find other employment to support his family. Despite overtures from the Academy in St Petersburg, which was seeking a successor to Euler, he chose instead to accept a position as director of the Observatory in Göttingen, a small university town in Lower Saxony. He spent his time tracking more asteroids through the night sky and completing surveys of the land for the Hanoverian and Danish governments. But he was always thinking about mathematics. Whilst charting the mountains of Hanover he would ponder Euclid’s axiom of parallel lines, and back in the observatory he would continue to expand his table of primes. Gauss had heard the first big theme in the music of the primes. But it was one of his few students, Riemann, who would truly unleash the full force of the hidden harmonies that lay behind the cacophony of the primes.

CHAPTER THREE
Riemann’s Imaginary Mathematical Looking-Glass

Do you not feel and hear it? Do I alone hear this melody so wondrously and gently sounding … Richard Wagner. Tristan und Isolde (Act III, Scene iii)

In 1809 Wilhelm von Humboldt became the education minister for the north German state of Prussia. In a letter to Goethe in 1816 he wrote, ‘I have busied myself here with science a great deal, but I have deeply felt the power antiquity has always wielded over me. The new disgusts me …’ Humboldt favoured a movement away from science as a means to an end, and a return to a more classical tradition of the pursuit of knowledge for its own sake. Previous education schemes had been geared to providing civil servants for the greater glory of Prussia. From now on, more emphasis was to be placed on education serving the needs of the individual rather than the state.

In his role as a thinker and civil servant, Humboldt enacted a revolution with far-reaching effects. New schools, called Gymnasiums, were created across Prussia and the neighbouring state of Hanover. Eventually the teachers in these schools were not to be members of the clergy, as in the old education system, but graduates of the new universities and polytechnics that were built during this period.

The jewel in the crown was Berlin University, founded in 1810 during the French occupation. Humboldt called it the ‘mother of all modern universities’. Housed in what had once been the palace of Prince Heinrich of Prussia on the grand boulevard Unter den Linden, the university would for the first time promote research alongside teaching. ‘University teaching not only enables an understanding of the unity of science but also its furtherance,’ Humboldt declared. Despite his passion for the Ancient World, under his guidance the university pioneered the introduction of new disciplines to sit beside the classical faculties of Law, Medicine, Philosophy and Theology.

For the first time, the study of mathematics was to form a major part of the curriculum in the new Gymnasiums and universities. Students were encouraged to study mathematics for its own sake and not simply as a servant of the other sciences. This contrasted starkly with Napoleon’s educational reforms, which saw mathematics harnessed to further French military aims. Carl Jacobi, one of the professors in Berlin, wrote to Legendre in Paris in 1830 about the French mathematician Joseph Fourier, who had reproached the German school of thought for ignoring more practical problems:

It is true that Fourier was of the opinion that the principal object of mathematics is public use and the explanation of natural phenomena; but a philosopher like him ought to have known that the sole object of the science is the honour of the human spirit, and that on this view a problem in the theory of numbers is worth as much as a problem of the system of the world.

For Napoleon, it was education that would finally destroy the arcane rules of the ancien régime. His recognition that education was the backbone for building his new France had led to the establishment in Paris of some of the institutes which are still famous today. Not only were the colleges meritocratic, allowing students from all backgrounds to attend, but also the educational philosophy put a greater emphasis on education and science serving society. One of the French Revolutionary regional officers wrote to a professor of mathematics in 1794, commending him on teaching a course in ‘Republican arithmetic’: ‘Citizen. The Revolution not only improves our morals and paves the way for our happiness and that of future generations, it even unlooses the shackles that hold back scientific progress.’

Humboldt’s approach to mathematics was very different from this utilitarian philosophy that prevailed across the border. The liberating effect of Germany’s educational revolution was to have a great impact on mathematicians’ understanding of many aspects of their field. It would allow them to establish a new, more abstract language of mathematics. In particular, it would revolutionise the study of prime numbers.

One town that benefited from Humboldt’s initiatives was Lüneberg, in Hanover. Lüneberg, once a thriving commercial centre, was now a town in decline. Its narrow streets paved with cobblestones were no longer buzzing with the business it had seen in previous centuries. But in 1829 a new building was erected amidst the tall towers of the three Gothic churches in Lüneberg: the Gymnasium Johanneum.

By the early 1840s the new school was flourishing. Its director, Schmalfuss, was a keen proponent of the neo-humanist ideals initiated by Humboldt. His library reflected his enlightened views: it featured not only the classics and the works of modern German writers, but also volumes from farther afield. In particular, Schmalfuss managed to get his hands on books coming out of Paris, the powerhouse of European intellectual activity during the first half of the century.

Schmalfuss had just accepted a new boy at the Gymnasium Johanneum, Bernhard Riemann. Riemann was very shy and found it difficult to make friends. He had been attending the Gymnasium in the town of Hanover, where he had been boarding with his grandmother, but when she died, in 1842, he was forced to move to Lüneberg where he could board with one of the teachers. Joining the school after all his contemporaries had established friendships did not make life easy for Riemann. He was desperately homesick and was teased by the other children. He would rather walk the long distance back to his father’s house in Quickborn than play with his contemporaries.

Riemann’s father, the pastor in Quickborn, had high expectations for his son. Although Bernhard was unhappy at school, he worked hard and conscientiously, determined not to disappoint his father. But he had to battle with an almost disabling streak of perfectionism. His teachers would often get frustrated at Riemann’s inability to submit his work. Unless it was perfect, the boy could not bear to suffer the indignity of anything less than full marks. His teachers began to doubt whether Riemann would ever be able to pass his final examinations.

It was Schmalfuss who saw a way to bring the young boy on and exploit his obsession with perfection. Early on, Schmalfuss had spotted Riemann’s special mathematical skills and was keen to stimulate the student’s abilities. He allowed Riemann the freedom of his library, with its fine collection of books on mathematics, where the boy could escape the social pressures of his classmates. The library opened up a whole new world for Riemann, a place where he felt at home and in control. Suddenly here he was in a perfect, idealised mathematical world where proof prevented any collapse of this new world around him, and numbers became his friends.

Humboldt’s drive from teaching science as a practical tool to the more aesthetic notion of knowledge for its own sake had filtered down to Schmalfuss’s classroom. The teacher steered Riemann’s reading away from mathematical texts full of formulas and rules that were aimed at feeding the demands of a growing industrial world, and guided him towards the classics of Euclid, Archimedes and Apollonius. With their geometry, the ancient Greeks sought to understand the abstract structure of points and lines, and they were not hung up on the particular formulas behind the geometry. When Schmalfuss did give Riemann a more modern text, Descartes’s treatise on analytical geometry – a subject rife with equations and formulas – the teacher could see that the mechanical method developed in the book did not appeal to Riemann’s growing taste for conceptual mathematics. As Schmalfuss later recalled in a letter to a friend, ‘already at that time he was a mathematician next to whose wealth a teacher felt poor’.

One of the books that was sitting on Schmalfuss’s shelf was a contemporary volume the teacher had acquired from France. Published in 1808, Théorie des Nombres by Adrien-Marie Legendre was the first text to record the observation that there seemed to be a strange connection between the function that counted the number of prime numbers and the logarithm function. This connection, discovered by Gauss and Legendre, was only based on experimental evidence. It was far from clear whether, as one counted higher, the number of primes would always be approximated by Gauss’s or Legendre’s function.

Despite the volume’s 859 large quarto pages, Riemann gobbled it up, returning the book to his teacher just six days later with the precocious declaration, ‘This is a wonderful book; I know it by heart.’ His teacher could not believe it, but when he examined Riemann on its contents during his final examinations two years later, the student answered perfectly. That marked the beginning of the career of one of the giants of modern mathematics. Thanks to Legendre, a seed was sown in the young Riemann’s mind that in later life was to blossom in spectacular fashion.

His final examinations over, Riemann was eager to join one of the vigorous new universities that were driving the educational revolution in Germany. His father, though, had other ideas. Riemann’s family was poor, and his father hoped that Bernhard would join him in the Church. The life of a clergyman would provide him with a regular income with which he could support his sisters. The only university in the Kingdom of Hanover to offer theology was not one of these new establishments but the University of Göttingen, founded over a century before, in 1734. So in 1846, to comply with his father’s wishes, Riemann made his way to the dank town of Göttingen.

Göttingen sits quietly in the gentle hills of Lower Saxony. At its heart lies a small medieval town enclosed by ancient ramparts. This is the Göttingen that Riemann would have known, and it still retains much of its original character. The streets wind narrowly between half-timbered houses topped with red-tiled roofs. The Brothers Grimm wrote many of their fairy tales in Göttingen, and one can imagine Hansel and Gretel running through its streets. In the centre stands the medieval town hall, whose walls display the motto ‘Away from Göttingen there is no life.’ For those at the university that was certainly the feeling. The academic life was one of self-sufficiency. Although theology had predominated in the early years of the university, the winds of academic change sweeping across Germany had stimulated Göttingen’s scientific curriculum. By the time Gauss was appointed as professor of astronomy and director of the observatory in 1807, it was science rather than theology for which Göttingen was becoming famous.

The fire for mathematics that the teacher Schmalfuss had ignited in the young Riemann was still burning strong. His father’s wish that he study theology had brought him to Göttingen, but it was the influence of the great Gauss and Göttingen’s scientific tradition that left its mark during that first year. It was only a matter of time before Greek and Latin lectures gave way to the temptation of courses in physics and mathematics. With trepidation, Riemann wrote to his father suggesting that he would like to switch from theology to mathematics. His father’s approval meant everything to Riemann. With a sense of relief he received his father’s blessing, and immediately immersed himself in the scientific life of the university.

To a young man of such talent, Göttingen soon began to feel small. Within a year Riemann had exhausted the resources available to him. Gauss, by now an old man, had become quite withdrawn from the intellectual life of the university – since 1828 he had spent only one night away from the observatory, where he lived. At the university he only lectured on astronomy, and in particular on the method that had made him famous when he’d rediscovered the ‘lost’ planet Ceres many years before. Riemann had to look elsewhere for the stimulus he needed to take the next step in his development. He could see Berlin was where the buzz of intellectual activity was the loudest.

The University of Berlin had been greatly influenced by the successful French research institutes, such as the École Polytechnique, that had been founded by Napoleon. It had, after all, been founded during the French occupation. One of the key mathematical ambassadors was a brilliant mathematician by the name of Peter Gustav Lejeune-Dirichlet. Although he was born in Germany in 1805, Dirichlet’s family was of French origin. A return to his roots took him to Paris in 1822, where he spent five years soaking up the intellectual activity that was bubbling out of the academies. Wilhelm von Humboldt’s brother Alexander, an amateur scientist, met Dirichlet on his travels and was so impressed that he secured him a position back in Germany. Dirichlet was something of a rebel. Perhaps the atmosphere on the streets in Paris had given him a taste for challenging authority. In Berlin, he was quite happy to ignore some of the antiquated traditions demanded by the rather stuffy university authorities, and often flouted their demands to demonstrate his command of the Latin language.

Göttingen and Berlin offered emerging scientists such as Riemann contrasting academic climates. Göttingen revelled in its independence and isolation. Few seminars were presented by anyone from beyond the city walls. It was self-sufficient and generated great science from the fuel burning within. Berlin, on the other hand, thrived on the stimulation coming from outside Germany. The ideas feeding through from France mixed with the forward-looking German approach to natural philosophy to create a heady new cocktail.

The different climates of Göttingen and Berlin suited different mathematicians. Some would never have succeeded without contact with new ideas from external sources. The success of other mathematicians can be traced back to an isolation which forced them to find an inner strength and new languages and modes of thought. Riemann would turn out to be someone whose breakthroughs came from contact with the wealth of new ideas that were in the air, and he could see that Berlin was the place to be.

Riemann made his move in 1847 and remained in Berlin for two years. While there, he was able to get his hands on papers by Gauss which he had not been able to prise from the reticent master in Göttingen. He attended lectures by Dirichlet, who was later to play a part in Riemann’s dramatic discoveries about prime numbers. By all accounts, Dirichlet was an inspiring lecturer. One mathematician who attended his lectures put it thus:

Dirichlet cannot be surpassed for richness of material and clear insight … he sits at the high desk facing us, puts his spectacles up on his forehead, leans his head on both hands, and … inside his hands he sees an imaginary calculation and reads it out to us – that we understand it as well as if we too saw it. I like that kind of lecturing.

Riemann made friends with several young researchers in Dirichlet’s seminars who were equally fired up by their passion for mathematics.

Other forces were also bubbling away in Berlin. The revolution of 1848 that swept away the French monarchy spread from the streets of Paris throughout much of Europe. It found its way to the streets of Berlin while Riemann was studying there. According to accounts of his contemporaries, it had a profound impact on him. On one of the few occasions in his life on which he joined with those around him on anything other than an intellectual level, he enlisted in the student corps defending the king in his Berlin palace. It is reputed that he did a continuous sixteen-hour stint on the barricades.

Riemann’s response to the mathematical revolution spreading from the Paris academies was not that of a reactionary. Berlin was importing not only political propaganda from Paris, but also many of the prestigious journals and publications coming out of the academies. Riemann received the latest volumes of the influential French journal Comptes Rendus and holed himself up in his room to pore over papers by the mathematical revolutionary Augustin-Louis Cauchy.

Cauchy was a child of the Revolution, born a few weeks after the fall of the Bastille in 1789. Undernourished by the little food available during those years, the feeble young Cauchy preferred to exercise his mind rather than his body. In time-honoured fashion, the mathematical world provided a refuge for him. A mathematical friend of Cauchy’s father, Lagrange, recognised the young boy’s precocious talent and commented to a contemporary, ‘You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.’ But he had interesting advice for Cauchy’s father. ‘Don’t let him touch a mathematical book till he is seventeen.’ Instead, he suggested stimulating the boy’s literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.

It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened. Cauchy’s output grew to be so immense that the journal Comptes Rendus had to impose a page limit on articles it printed that is strictly adhered to even today. Cauchy’s new mathematical language was too much for some of his contemporaries. The Norwegian mathematician Niels Henrik Abel wrote in 1826, ‘Cauchy is mad … what he does is excellent but very muddled. At first I understood practically none of it; now I see some of it more clearly.’ Abel goes on to observe that of all the mathematicians in Paris, Cauchy was the only one doing ‘pure mathematics’ whilst others ‘busy themselves exclusively with magnetism and other physical subjects … he is the only one who knows how mathematics should be done’.

Cauchy was to land himself in trouble with the authorities in Paris for steering students away from practical applications of mathematics. The director of the École Polytechnique, where Cauchy was lecturing, wrote to him criticising him for his obsession with abstract mathematics: ‘It is the opinion of many persons that instruction in pure mathematics is being carried too far at the École and that such an uncalled for extravagance is prejudicial to the other branches.’ So it was perhaps no wonder that Cauchy’s work would be appreciated by the young Riemann.

So exciting were these new ideas that Riemann almost became a recluse. His contemporaries were to see nothing of him while he waded through Cauchy’s output. Several weeks later Riemann resurfaced, declaring that ‘this is a new mathematics’. What had captured Cauchy and Riemann’s imagination was the emerging power of imaginary numbers.

Imaginary numbers – a new mathematical vista

The square root of minus one, the building block of imaginary numbers, seems to be a contradiction in terms. Some say that admitting the possibility of such a number is what separates the mathematicians from the rest. A creative leap is required to gain access to this bit of the mathematical world. At first sight it looks as if it has nothing to do with the physical world. The physical world seems to be built on numbers whose square is always a positive number. Imaginary numbers, however, are more than just an abstract game. They hold the key to the twentieth-century world of subatomic particles. On a larger scale, aeroplanes would not have taken to the skies without engineers taking a journey through the world of imaginary numbers. This new world provides a flexibility denied to those who stick to ordinary numbers.

The story of how these new numbers were discovered begins with the need to solve simple equations. As the ancient Babylonians and Egyptians recognised, if seven fish were to be divided between three people, for example, fractional numbers – , and so on – would have to come into the equation. By the sixth century BC, the Greeks had discovered while exploring the geometry of triangles that these fractions were sometimes incapable of expressing the lengths of the sides of a triangle. Pythagoras’ theorem forced them to invent new numbers that couldn’t be written as simple fractions. For example, Pythagoras could take a right-angled triangle whose two shortest sides are one unit long. His famous theorem then told him that the longest side had length x, where x is a solution of the equation x² = 1² + 1² = 2. In other words, the length is the square root of 2.

Fractions are the numbers whose decimal expansions have a repeating pattern. For example, In contrast, the Greeks could prove that the square root of 2 is not equal to a fraction. However far you calculate the decimal expansion of the square root of 2, it will never settle down into such a repeating pattern. The square root of 2 starts off 1.414213562 … Riemann used to idle away the hours calculating more and more of these decimal places during his years in Göttingen. His record was thirty-eight places, no mean feat without a computer but perhaps more a reflection on the dull Göttingen nightlife and Riemann’s shy persona that this was his evening entertainment. Nonetheless, however far Riemann calculated, he knew that he could never write down the complete number or discover a repeating pattern.

To capture the impossibility of expressing such numbers in any way other than as solutions to equations such as x² = 2, mathematicians called them irrational numbers. The name reflected mathematicians’ sense of unease at their inability to write down precisely what these numbers were. Nevertheless, there was still a sense of the reality of these numbers since they could be seen as points marked on a ruler, or on what mathematicians call the number line. The square root of 2, for example, is a point somewhere between 1.4 and 1.5. If one could make a perfect Pythagorean right-angled triangle with the two short sides one unit long, then the location of this irrational number could be determined by laying the long side against the ruler and marking off the length.

The negative numbers were discovered similarly out of attempts to solve simple equations such as x + 3 = 1. Hindu mathematicians proposed these new numbers in the seventh century AD. Negative numbers were created in response to the growing world of finance, as they were useful for describing debt. It took European mathematicians another millennium before they were happy to admit the existence of such ‘fictitious numbers’, as they were called. Negative numbers took their place on the number line stretching out to the left of zero.

The real numbers – every fraction, negative number or irrational number is represented by a point on the number line.

Irrational numbers and negative numbers allow us to solve many different equations. Fermat’s equation x³ + y³ = z³ has interesting solutions if you don’t insist, as Fermat had, that x, y and z should be whole numbers. For example, we can choose x = 1 and y = 1, and put z equal to the cube root of 2 – and the equation is solved. But there were still other equations which couldn’t be solved in terms of any of the numbers on the number line.

There seemed to be no number which was a solution to the equation x² = −1. After all, if you square a number, positive or negative, the answer is always positive. So any number that satisfies this equation is not going to be an ordinary number. But if the Greeks could imagine a number like the square root of 2, without being able to write it down as a fraction, mathematicians began to see that they could make a similar imaginative leap and create a new number to solve the equation x² = −1. Such a creative jump marks one of the conceptual challenges for anyone learning mathematics. This new number, the square root of minus one, was called an imaginary number and given the symbol i. In contrast, mathematicians began to refer to the numbers that could be found on the number line as real numbers.

To create an answer to this equation, seemingly out of thin air, seems like cheating. Why not accept that the equation has no solution? That is one way forward, but mathematicians like to be more optimistic. Once we accept the idea that there is a new number that does solve this equation, the advantages of this creative step far outweigh any initial unease. Once named, its existence seems inevitable. It no longer feels like an artificially created number but a number that had been there all along, unobserved until we’d asked the right question. Eighteenth-century mathematicians had been loath to admit there could be any such numbers. Nineteenth-century mathematicians were brave enough to believe in new modes of thought which challenged the accepted ideas of what constituted the mathematical canon.

Frankly, the square root of −1 is as abstract a concept as the square root of 2. Both are defined as solutions to equations. But would mathematicians have to start creating new numbers for every new equation that came along? What if we want solutions to an equation like x⁴ = −1? Are we going to have to use more and more letters in our attempts to name all these new solutions? It was with some relief that Gauss finally proved, in his doctoral thesis of 1799, that no new numbers were needed. Using this new number i, mathematicians could finally solve any equation they might come across. Every equation had a solution that consisted of some combination of ordinary real numbers (the fractions and irrational numbers) and this new number, i.

The key to Gauss’s proof was to extend the picture we already had of ordinary numbers as lying on a number line: a line running east-west on which each point represents a number. These were the real numbers familiar to mathematicians since the Greeks. But there was no room on the line for this new imaginary number, the square root of −1. So Gauss wondered what would happen if you created a new direction. What if one unit north of the number line were used to represent i? All the new numbers that were needed to solve equations were combinations of i and ordinary numbers, for example 1 + 2i. Gauss realised that on this two-dimensional map there was a point corresponding to every possible number. The imaginary numbers then simply became coordinates on a map. The number 1 + 2i was represented by the point reached by travelling one unit east and two units north.

Gauss would interpret these numbers as sets of directions in his map of the imaginary world. To add two imaginary numbers A + Bi and C + Di just meant following two sets of directions, one after the other. For example, adding together 6 + 3i and 1 + 2i gets you to the location 7 + 5i.

Following directions – how to add two imaginary numbers.

Although this was a very potent picture. Gauss was to keep his map of the imaginary world hidden from public view. Once he had built his proof, he removed the graphic scaffolding so that no trace of his vision remained. He was aware that pictures in mathematics were regarded with some suspicion during this period. The dominance of the French mathematical tradition during Gauss’s youth meant that the preferred pathway to the mathematical world was the language of formulas and equations, a language that went hand in hand with the utilitarian approach to the subject. There were also other reasons for this aversion to images.

For several hundred years, mathematicians had believed that pictures had the power to mislead. After all, the language of mathematics had been introduced to tame the physical world. In the seventeenth century, Descartes had attempted to turn the study of geometry into pure statements about numbers and equations. ‘Sense perceptions are sense deceptions’ was his motto. Riemann had come to dislike this denial of the physical picture when he’d been reading Descartes in the comfort of Schmalfuss’s library.

Mathematicians around the turn of the nineteenth century had been burnt by an erroneous pictorial proof of a formula describing the relationship between the number of corners, edges and faces of geometric solids. Euler had conjectured that if a solid has C corners, E edges and F faces, then the numbers C, E and F must satisfy the relationship C − E + F = 2. For example, a cube has 8 corners, 12 edges and 6 faces. The young Cauchy had himself constructed a ‘proof’ in 1811 based on pictorial intuition, but was rather shocked to be shown a solid which didn’t satisfy this formula – a cube with a hole at its centre.

The ‘proof’ had missed the possibility that a solid might contain such a hole. It was necessary to introduce an extra ingredient into the formula which kept track of the number of holes in the solid. Having been tricked by the power of pictures to hide perspectives that weren’t initially apparent, Cauchy sought refuge in the security that formulas seemed to provide. One of the revolutions he effected was to create a new mathematical language which allowed mathematicians to discuss the concept of symmetry in a rigorous way without the need for pictures.

Gauss knew that his secret map of imaginary numbers would be anathema to mathematicians at the end of the eighteenth century, so he omitted it from his proof. Numbers were things you added and multiplied, not drew pictures of. Gauss eventually came clean some forty years later about the scaffolding he had used in his doctorate.

Looking-glass world

Even without Gauss’s map, Cauchy and other mathematicians had begun to explore what happens if you extend the idea of functions to this new world of imaginary numbers rather than sticking to real numbers. To their surprise, these imaginary numbers opened up new connections between seemingly unrelated parts of the mathematical world.

A function is like a computer program in which you input one number, a calculation is made, and another number is output. The function might be defined by some simple equation like x² + 1. When you input a number, for example 2, the function calculates 2² + 1 and outputs 5. Other functions are more complicated. Gauss was interested in the function that counted the number of primes. You input a number x, and the function tells you how many primes there are up to x. Gauss had denoted this function by π(x). The graph of this function is a climbing staircase, as shown on p. 50. Every time the input encounters a prime number, the output jumps another step. As x goes from 4.9 to 5.1, the number of primes increases from two to three to register the new prime, 5.

Mathematicians soon realised that some of these functions, such as the one built from the equation x² + 1, could be fed with imaginary numbers as well as ordinary numbers. For example, input x = 2i into the function, and the output is calculated as (2i)² + 1 = −4 + 1 = −3. The feeding of functions with imaginary numbers had begun in Euler’s generation. As early as 1748, Euler had stumbled across strange connections between unrelated bits of mathematics by taking a trip through this looking-glass world. Euler knew that when you fed the exponential function 2x with ordinary numbers x, you got a graph that climbed rapidly. But when he fed the function with imaginary numbers he got a rather unexpected answer. Instead of the exponentially climbing graph, he started to see undulating waves of the type we now associate with, for example, sound waves. The function that produces these undulating waves is called the sine function. The image of the sine function is a familiar repeating curve where every 360 degrees we see the same shape appearing. The sine function is now used in a host of everyday calculations. For example, it can be used to measure the height of a building from ground level by measuring angles. It was Euler’s generation who discovered that these sine waves were also the key to reproducing musical sounds. A pure note like the A produced by a tuning fork used to tune a piano can be represented by such a wave.

Euler fed imaginary numbers into the function 2x. To his surprise, out came waves which corresponded to a particular musical note. Euler showed that the character of each note depended on the coordinates of the corresponding imaginary number. The farther north one is, the higher the pitch. The farther east, the louder the volume. Euler’s discovery was the first inkling that these imaginary numbers might open up unexpected new paths through the mathematical landscape. Following Euler, mathematicians began travelling out to this new-found land of imaginary numbers. The search for new connections would become infectious.

Riemann returned to Göttingen in 1849 in order to complete his doctoral thesis for Gauss’s consideration. That was the year in which Gauss wrote to his friend Encke of his childhood discovery of the connection between primes and logarithms. Although Gauss probably discussed his discovery with members of the faculty in Göttingen, prime numbers were not yet on Riemann’s mind. He was buzzing with the new mathematics from Paris, keen to explore the strange emerging world of functions fed with imaginary numbers.

Cauchy had begun the task of making a rigorous subject out of Euler’s first tentative steps into this new territory. Whilst the French were masters of equations and formulaic manipulation, Riemann was ready to capitalise on the German education system’s return to a more conceptual view of the world. By November 1851, his ideas had crystallised and he submitted his dissertation to the faculty in Göttingen. His ideas obviously struck a chord with Gauss. He greeted Riemann’s doctorate as evidence ‘of a creative, active, truly mathematical mind, and of a gloriously fertile originality’.

Riemann wrote to his father, keen to tell him about the progress he was making: ‘I believe I have improved my prospects with my dissertation. I hope also to learn to write more quickly and more fluently in time, especially if I mingle in society.’ But academic life in Göttingen did not at first live up to the thrills of Berlin. It was a somewhat stuffy, insular university, and Riemann lacked the confidence to engage with the old intellectual hierarchy. There were fewer students in Göttingen with whom he could relate. He was mistrustful of other people and never really at ease in a social environment. ‘He has done the strangest things here only because he believes that nobody can bear him,’ wrote his contemporary Richard Dedekind. Riemann was a hypochondriac and susceptible to bouts of depression. He hid his face behind the security of an increasingly large black beard. He was extremely anxious about his finances, surviving on the uncertainty of half a dozen voluntary students’ fees. The workload he undertook combined with the pressures of poverty led to a temporary breakdown in 1854. However, his mood would lighten whenever Dirichlet, the star of the Berlin mathematical tradition, visited Göttingen.

One professor in Göttingen with whom Riemann did manage to strike up a friendship was the eminent physicist Wilhelm Weber. Weber had collaborated on numerous projects with Gauss during their time together in Göttingen. They became a scientific Sherlock Holmes and Dr Watson, Gauss providing the theoretical underpinning which Weber then put into practice. One of their most famous inventions was to realise the potential of electromagnetism to communicate over a distance. They successfully rigged up a telegraph line between Gauss’s observatory and Weber’s laboratory over which they exchanged messages.

Although Gauss thought the invention a curiosity, Weber saw clearly what their discovery would unleash. ‘When the globe is covered with a net of railroads and telegraph wires,’ he wrote, ‘this net will render services comparable to those of the nervous system in the human body, partly as a means of transport, partly as a means for the propagation of ideas and sensations with the speed of lightning.’ The rapid spread of the telegraph, together with the later implementation of Gauss’s invention of the clock calculator in computer security, make Gauss and Weber the grandfathers of e-business and the Internet. Their collaboration is immortalised in a statue of the pair in the city of Göttingen.

One visitor to Weber in Göttingen paints a typical picture of a slightly mad inventor: ‘a curious little fellow who speaks in a shrill, unpleasant and hesitating voice. He speaks and stutters unceasingly; one has nothing to do but to listen. Sometimes he laughs for no earthly reason, and one feels sorry at not being able to join him.’ Weber had a little more of the rebel in him than his collaborator Gauss. He had been one of the ‘Göttingen Seven’ temporarily dismissed from the faculty for protesting at the arbitrary rule of the Hanoverian king in 1837. For some time after completing his thesis, Riemann worked as Weber’s assistant. During this apprenticeship Riemann developed a soft spot for Weber’s daughter, but his advances were not reciprocated.

In 1854, Riemann wrote to his father that ‘Gauss is seriously ill and the physicians fear that his death is imminent.’ Riemann was worried that Gauss might die before examining his habilitation, the degree required to become a professor at a German university. Fortunately Gauss lived long enough to hear Riemann’s ideas on geometry and its relationship to physics that had germinated during his work with Weber. Riemann was convinced that the fundamental questions of physics could all be answered using mathematics alone. The developments in physics over the ensuing years would eventually confirm his faith in mathematics. Riemann’s theory of geometry is regarded by many as one of his most significant contributions to science, and it would be one of the planks in the platform from which Einstein launched his scientific revolution at the beginning of the twentieth century.

A year later, Gauss died. Although the man had passed, his ideas were to keep mathematicians busy for generations to come. He had left behind his conjectured connection between primes and the logarithm function for the next generation to chew over. Astronomers immortalised the great man in the heavens by naming an asteroid Gaussia. And in the University of Göttingen’s anatomical collection one can even find Gauss’s brain, pickled for eternity, which was reported to be more richly convoluted than any brain previously dissected.

Dirichlet, whose lectures Riemann had attended in Berlin, was appointed to Gauss’s vacant chair. Dirichlet was to bring to Göttingen some of the intellectual excitement that Riemann had enjoyed when he was in Berlin. An English mathematician recorded the impression that a visit to Dirichlet in Göttingen made on him at this time: ‘He is rather tall, lanky-looking man with moustache and beard about to turn grey … with a somewhat harsh voice and rather deaf: it was early, he was unwashed and unshaved and with his schlafrock [dressing gown], slippers, cup of coffee and cigar.’ Despite this Bohemian exterior, there burned inside him a desire for rigour and proof that was unequalled at the time. His contemporary in Berlin, Carl Jacobi, wrote to Dirichlet’s first patron Alexander von Humboldt that ‘Only Dirichlet, not I, nor Cauchy, not Gauss, knows what a perfectly rigorous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain.’

The arrival of Dirichlet in Göttingen began to shake the social fabric of the town. Dirichlet’s wife, Rebecka, was the sister of the composer Felix Mendelssohn. Rebecka loathed the dull Göttingen social scene and threw numerous parties trying to reproduce the Berlin salon atmosphere she had been forced to leave behind.

Dirichlet’s less formal approach to the educational hierarchy meant that Riemann was able to discuss mathematics openly with the new professor. Riemann had become rather isolated on his return from Berlin to Göttingen. The combination of Gauss’s austere personality in later life and Riemann’s shyness meant that Riemann had discussed little with the great master. By contrast, Dirichlet’s relaxed manner was perfect for Riemann who, in an atmosphere more conducive to discussion, began to open up. Riemann wrote to his father about his new mentor: ‘Next morning Dirichlet was with me for two hours. He read over my dissertation and was very friendly – which I could hardly have expected considering the great distance in rank between us.’

In turn, Dirichlet appreciated Riemann’s modesty and also recognised the originality of his work. On occasions Dirichlet even managed to drag Riemann away from the library to join him on walks in the countryside around Göttingen. Almost apologetically, Riemann wrote to his father explaining that these escapes from mathematics did him more good scientifically than if he had stayed at home poring over his books. It was during one of his discussions with Riemann whilst walking through the woods of Lower Saxony that Dirichlet inspired Riemann’s next move. It would open up a whole new perspective on the primes.

The zeta function – the dialogue between music and mathematics

During his years in Paris in the 1820s, Dirichlet had become fascinated by Gauss’s great youthful treatise Disquisitiones Arithmeticae. Although Gauss’s book marked a beginning of number theory as an independent discipline, the book was difficult and many failed to penetrate the concise style Gauss preferred. Dirichlet, though, was more than happy to battle with one tough paragraph after another. At night he would place the book under his pillow in the hope that the next morning’s reading would suddenly make sense. Gauss’s treatise has been described as a ‘book of seven seals’, but thanks to the labours and dreams of Dirichlet, those seals were broken and the treasures within gained the wide distribution they deserved.

Dirichlet was especially interested in Gauss’s clock calculator. In particular, he was intrigued by a conjecture that went back to a pattern spotted by Fermat. If you took a clock calculator with N hours on it and you fed in the primes, then, Fermat conjectured, infinitely often the clock would hit one o’clock. So, for example, if you take a clock with 4 hours there are infinitely many primes which Fermat predicted would leave remainder 1 on division by 4. The list begins 5, 13, 17, 29, …

In 1838, at the age of thirty-three, Dirichlet had made his mark in the theory of numbers by proving that Fermat’s hunch was indeed correct. He did this by mixing ideas from several areas of mathematics that didn’t look as if they had anything to do with one another. Instead of an elementary argument like Euclid’s cunning proof that there are infinitely many primes, Dirichlet used a sophisticated function that had first appeared on the mathematical circuit in Euler’s day. It was called the zeta function, and was denoted by the Greek letter ζ. The following equation provided Dirichlet with the rule for calculating the value of the zeta function when fed with a number x:

To calculate the output at x, Dirichlet needed to carry out three mathematical steps. First, calculate the exponential numbers 1x, 2x, 3x, …, nx, … Then take the reciprocals of all the numbers produced in the first step. (The reciprocal of 2x is 1/2x.) Finally, add together all the answers from the second step.

It is a complicated recipe. The fact that each number 1, 2, 3, … makes a contribution to the definition of the zeta function hints at its usefulness to the number theorist. The downside comes in having to deal with an infinite sum of numbers. Few mathematicians could have predicted what a powerful tool this function would become as the best way to study the primes. It was almost stumbled upon by accident.

The origins of mathematicians’ interest in this infinite sum came from music and went back to a discovery made by the Greeks. Pythagoras was the first to discover the fundamental connection between mathematics and music. He filled an urn with water and banged it with a hammer to produce a note. If he removed half the water and banged the urn again, the note had gone up an octave. Each time he removed more water to leave the urn one-third full, then one-quarter full, the notes produced would sound to his ear in harmony with the first note he’d played. Any other notes which were created by removing some other amount of water sounded in dissonance with that original note. There was some audible beauty associated with these fractions. The harmony that Pythagoras had discovered in the numbers made him believe that the whole universe was controlled by music, which is why he coined the expression ‘the music of the spheres’.

Ever since Pythagoras’ discovery of an arithmetic connection between mathematics and music, people have compared both the aesthetic and the physical traits shared by the two disciplines. The French Baroque composer Jean-Philippe Rameau wrote in 1722 that ‘Not withstanding all the experience I may have acquired in music from being associated with it for so long, I must confess that only with the aid of mathematics did my ideas become clear.’ Euler sought to make music theory ‘part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing’. Euler believed that it was the primes that lay behind the beauty of certain combinations of notes.

Many mathematicians have a natural affinity with music. Euler would relax after a hard day’s calculating by playing his clavier. Mathematics departments invariably have little trouble assembling an orchestra from the ranks of their members. There is an obvious numerical connection between the two given that counting underpins both. As Leibniz described it, ‘Music is the pleasure the human mind experiences from counting without being aware that it is counting.’ But the resonance between the subjects goes much deeper than this.

Mathematics is an aesthetic discipline where talk of beautiful proofs and elegant solutions is commonplace. Only those with a special aesthetic sensibility are equipped to make mathematical discoveries. The flash of illumination that mathematicians crave often feels like bashing notes on a piano until suddenly a combination is found which contains an inner harmony marking it out as different.

G.H. Hardy wrote that he was ‘interested in mathematics only as a creative art’. Even for the French mathematicians in Napoleon’s academies, the buzz of doing mathematics came not from its practical application but from its inner beauty. The aesthetic experiences of doing mathematics or listening to music have much in common. Just as you might listen to a piece of music over and over and find new resonances previously missed, mathematicians often take pleasure in re-reading proofs in which the subtle nuances that make it hang together so effortlessly gradually reveal themselves. Hardy believed that the true test of a good mathematical proof was that ‘the ideas must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.’ For Hardy, ‘A mathematical proof should resemble a simple and clear-cut constellation, not a scattered Milky Way.’

Both mathematics and music have a technical language of symbols which allow us to articulate the patterns we are creating or discovering. Music is much more than just the minims and crochets which dance across the musical stave. Similarly, mathematical symbols come alive only when the mathematics is played with in the mind.

As Pythagoras discovered, it is not just in the aesthetic realm that mathematics and music overlap. The very physics of music has at its root the basics of mathematics. If you blow across the top of a bottle you hear a note. By blowing harder, and with a little skill, you can start to hear higher notes – the extra harmonics, the overtones. When a musician plays a note on an instrument they are producing an infinity of additional harmonics, just as you do when you blow across the top of the bottle. These additional harmonics help to give each instrument its own distinctive sound. The physical characteristics of each instrument mean that we hear different combinations of harmonics. In addition to the fundamental note, the clarinet plays only those harmonics produced by odd fractions: , … The string of a violin, on the other hand, vibrates to create all the harmonics that Pythagoras produced with his urn – those corresponding to the fractions , …

Since the sound of a vibrating violin string is the infinite sum of the fundamental note and all the possible harmonics, mathematicians became intrigued by the mathematical analogue. The infinite sum … became known as the harmonic series. This infinite sum is also the answer Euler got when he fed the zeta function with the number x = 1. Although this sum grew only very slowly as he added more terms, mathematicians had known since the fourteenth century that eventually it must spiral off to infinity.

So the zeta function must output the answer infinity when fed the number x = 1. If, however, instead of taking x = 1, Euler fed the zeta function with a number bigger than 1, the answer no longer spiralled off to infinity. For example, taking x = 2 means adding together all the squares in the harmonic series:

This is a smaller number as it does not include all possible fractions found when x = 1. We are now adding only some of the fractions, and Euler knew that this time the smaller sum wouldn’t spiral off to infinity but would home in on some particular number. It had become quite a challenge by Euler’s day to identify a precise value for this infinite sum when x = 2. The best estimate was somewhere around . In 1735, Euler wrote that ‘So much work has been done on the series that it seems hardly likely that anything new about them may still turn up … I, too, in spite of repeated effort, could achieve nothing more than approximate values for their sums.’

Nevertheless, Euler, emboldened by his previous discoveries, began to play around with this infinite sum. Twisting it this way and that like the sides of a Rubik’s cube, he suddenly found the series transformed. Like the colours on the cube, these numbers slowly came together to form a completely different pattern from the one he had started with. As he went on to describe, ‘Now, however, quite unexpectedly, I have found an elegant formula depending upon the quadrature of the circle’ – in modern parlance, a formula depending on the number π = 3.1415 …

By some pretty reckless analysis, Euler had discovered that this infinite sum was homing in on the square of π divided by 6:

The decimal expansion of , like that of π, is completely chaotic and unpredictable. To this day, Euler’s discovery of this order lurking in the number ranks as one of the most intriguing calculations in all of mathematics, and it took the scientific community of Euler’s time by storm. No one had predicted a link between the innocent sum and the chaotic number π.

This success inspired Euler to investigate the power of the zeta function further. He knew that if he fed the zeta function with any number bigger than 1, the result would be some finite number. After a few years of solitary study he managed to identify the output of the zeta function for every even number. But there was something rather unsatisfactory about the zeta function. Whenever Euler fed the formula for the zeta function with any number less than 1, it would always output infinity. For example, for x = −1 it yields the infinite sum 1 + 2 + 3 + 4 + … The function behaved well only for numbers bigger than 1.

Euler’s discovery of his expression for in terms of simple fractions was the first sign that the zeta function might reveal unexpected links between seemingly disparate parts of the mathematical canon. The second strange connection that Euler discovered was with an even more unpredictable sequence of numbers.

Rewriting the Greek story of the primes

Prime numbers suddenly enter Euler’s story as he tried to put his rickety analysis of the expression for on a sound mathematical footing. As he played with the infinite sums he recalled the Greek discovery that every number can be built from multiplying prime numbers together, and realised that there was an alternative way to write the zeta function. He spotted that every term in the harmonic series, for example , could be dissected using the knowledge that every number is built from its prime building blocks. So he wrote

Instead of writing the harmonic series as an infinite addition of all the fractions, Euler could take just fractions built from single primes, like , and multiply them together. His expression, known today as Euler’s product, connected the worlds of addition and multiplication. The zeta function appeared on one side of the new equation and the primes on the other. In one equation was encapsulated the fact that every number can be built by multiplying together prime numbers:

At first sight Euler’s product doesn’t look as if it will help us in our quest to understand prime numbers. After all, it’s just another way of expressing something the Greeks knew more than two thousand years ago. Indeed, Euler himself would not grasp the full significance of his rewriting of this property of the primes.

The significance of Euler’s product took another hundred years, and the insight of Dirichlet and Riemann, to be recognised. By turning this Greek gem and staring at it from a nineteenth-century perspective, there emerged a new mathematical horizon that the Greeks could never have imagined. In Berlin, Dirichlet was intrigued by the way Euler had used the zeta function to express an important property of prime numbers – one that the Greeks had proved two thousand years before. When Euler input the number 1 into the zeta function, the output spiralled off to infinity. Euler saw that the output could spiral off to infinity only if there were infinitely many prime numbers. The key to this realisation was Euler’s product, which connected the zeta function and the primes. Although the Greeks had proved centuries before that there were infinitely many primes, Euler’s novel proof incorporated concepts completely different to those used by Euclid.

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