Kitabı oku: «The Music of the Primes: Why an unsolved problem in mathematics matters», sayfa 3

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 2^N 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 2^N 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.