Читать книгу The Music of the Primes: Why an unsolved problem in mathematics matters - Marcus Sautoy du - Страница 8

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.