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

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