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

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      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 СКАЧАТЬ