Fermat's Last Theorem Read online

  Life, Prince Leon, may well be compared with these public Games for in the vast crowd assembled here some are attracted by the acquisition of gain, others are led on by the hopes and ambitions of fame and glory. But among them there are a few who have come to observe and to understand all that passes here.

  It is the same with life. Some are influenced by the love of wealth while others are blindly led on by the mad fever for power and domination, but the finest type of man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher for although no man is completely wise in all respects, he can love wisdom as the key to nature’s secrets.

  Although many were aware of Pythagoras’ aspirations nobody outside of the Brotherhood knew the details or extent of his success. Each member of the school was forced to swear an oath never to reveal to the outside world any of their mathematical discoveries. Even after Pythagoras’ death a member of the Brotherhood was drowned for breaking his oath – he publicly announced the discovery of a new regular solid, the dodecahedron, constructed from twelve regular pentagons. The highly secretive nature of the Pythagorean Brotherhood is part of the reason that myths have developed surrounding the strange rituals which they might have practised, and similarly this is why there are so few reliable accounts of their mathematical achievements.

  What is known for certain is that Pythagoras established an ethos which changed the course of mathematics. The Brotherhood was effectively a religious community and one of the idols they worshipped was Number. By understanding the relationships between numbers, they believed that they could uncover the spiritual secrets of the universe and bring themselves closer to the gods. In particular the Brotherhood focused its attention on the study of counting numbers (1, 2, 3, …) and fractions. Counting numbers are sometimes called whole numbers, and together with fractions (ratios between whole numbers) are technically referred to as rational numbers. Among the infinity of numbers, the Brotherhood looked for those with special significance, and some of the most special were the so-called ‘perfect’ numbers.

  According to Pythagoras numerical perfection depended on a number’s divisors (numbers which will divide perfectly into the original one). For instance, the divisors of 12 are 1, 2, 3, 4 and 6. When the sum of a number’s divisors is greater than the number itself, it is called an ‘excessive’ number. Therefore 12 is an excessive number because its divisors add up to 16. On the other hand, when the sum of a number’s divisors is less than the number itself, it is called ‘defective’. So 10 is a defective number because its divisors (1, 2 and 5) add up to only 8.

  The most significant and rarest numbers are those whose divisors add up exactly to the number itself and these are the perfect numbers. The number 6 has the divisors 1, 2 and 3, and consequently it is a perfect number because 1 + 2 + 3 = 6. The next perfect number is 28, because 1 + 2 + 4 + 7 + 14 = 28.

  As well as having mathematical significance for the Brotherhood, the perfection of 6 and 28 was acknowledged by other cultures who observed that the moon orbits the earth every 28 days and who declared that God created the world in 6 days. In The City of God, St Augustine argues that although God could have created the world in an instant he decided to take six days in order to reflect the universe’s perfection. St Augustine observed that 6 was not perfect because God chose it, but rather that the perfection was inherent in the nature of the number: ‘6 is a number perfect in itself, and not because God created all things in six days; rather the inverse is true; God created all things in six days because this number is perfect. And it would remain perfect even if the work of the six days did not exist.’

  As the counting numbers get bigger the perfect numbers become harder to find. The third perfect number is 496, the fourth is 8,128, the fifth is 33,550,336 and the sixth is 8,589,869,056. As well as being the sum of their divisors, Pythagoras noted that all perfect numbers exhibit several other elegant properties. For example, perfect numbers are always the sum of a series of consecutive counting numbers. So we have

  Pythagoras was entertained by perfect numbers but he was not satisfied with merely collecting these special numbers; instead he desired to discover their deeper significance. One of his insights was that perfection was closely linked to ‘twoness’. The numbers 4 (2 × 2), 8 (2 × 2 × 2), 16 (2 × 2 × 2 × 2), etc., are known as powers of 2, and can be written as 2n, where the n represents the number of 2’s multiplied together. All these powers of 2 only just fail to be perfect, because the sum of their divisors always adds up to one less than the number itself. This makes them only slightly defective:

  Two centuries later Euclid would refine Pythagoras’ link between twoness and perfection. Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1. That is to say,

  Today’s computers have continued the search for perfect numbers and find such enormously large examples as 2216’090 × (2216’091 – 1), a number with over 130,000 digits, which obeys Euclid’s rule.

  Pythagoras was fascinated by the rich patterns and properties possessed by perfect numbers and respected their subtlety and cunning. At first sight perfection is a relatively simple concept to grasp and yet the ancient Greeks were unable to fathom some of the fundamental points of the subject. For example, although there are plenty of numbers whose divisors add up to one less than the number itself, that is to say only slightly defective, there appear to be no numbers which are slightly excessive. The Greeks were unable to find any numbers whose divisors added up to one more than the number itself, but they could not explain why this was the case. Frustratingly, although they failed to discover slightly excessive numbers, they could not prove that no such numbers existed. Understanding the apparent lack of slightly excessive numbers was of no practical value whatsoever; nonetheless it was a problem which might illuminate the nature of numbers and therefore it was worthy of study. Such riddles intrigued the Pythagorean Brotherhood, and two and a half thousand years later, mathematicians are still unable to prove that no slightly excessive numbers exist.

  Everything is Number

  In addition to studying the relationships within numbers Pythagoras was also intrigued by the link between numbers and nature. He realised that natural phenomena are governed by laws, and that these laws could be described by mathematical equations. One of the first links he discovered was the fundamental relationship between the harmony of music and the harmony of numbers.

  The most important instrument in early Hellenic music was the tetrachord or four-stringed lyre. Prior to Pythagoras, musicians appreciated that particular notes when sounded together created a pleasant effect, and tuned their lyres so that plucking two strings would generate such a harmony. However, the early musicians had no understanding of why particular notes were harmonious and had no objective system for tuning their instruments. Instead they tuned their lyres purely by ear until a state of harmony was established – a process which Plato called torturing the tuning pegs.

  Iamblichus, the fourth-century scholar who wrote nine books about the Pythagorean sect, decribes how Pythagoras came to discover the underlying principles of musical harmony:

  Once he was engrossed in the thought of whether he could devise a mechanical aid for the sense of hearing which would prove both certain and ingenious. Such an aid would be similar to the compasses, rules and optical instruments designed for the sense of sight. Likewise the sense of touch had scales and the concepts of weights and measures. By some divine stroke of luck he happened to walk past the forge of a blacksmith and listened to the hammers pounding iron and producing a variegated harmony of reverberations between them, except for one combination of sounds.

  According to Iamblichus, Pythagoras immediately ran into the forge to investigate the harmony of the hammers. He noticed that most of the hammers could be struck simultaneously to generate a harmonious sound, whereas any combination containin
g one particular hammer always generated an unpleasant noise. He analysed the hammers and realised that those which were harmonious with each other had a simple mathematical relationship – their masses were simple ratios or fractions of each other. That is to say that hammers half, two-thirds or three-quarters the weight of a particular hammer would all generate harmonious sounds. On the other hand, the hammer which was generating disharmony when struck along with any of the other hammers had a weight which bore no simple relationship to the other weights.

  Pythagoras had discovered that simple numerical ratios were responsible for harmony in music. Scientists have cast some doubt on Iamblichus’ account of this story, but what is more certain is how Pythagoras applied his new theory of musical ratios to the lyre by examining the properties of a single string. Simply plucking the string generates a standard note or tone which is produced by the entire length of the vibrating string. By fixing the string at particular points along its length, it is possible to generate other vibrations and tones. Crucially, harmonious tones only occur at very specific points. For example, by fixing the string at a point exactly half-way along it, plucking generates a tone which is one octave higher and in harmony with the original tone. Similarly, by fixing the string at points which are exactly a third, a quarter or a fifth of the way along it, other harmonious notes are produced. However, by fixing the string at a point which is not a simple fraction along the length of the whole string, a tone is generated which is not in harmony with the other tones.

  Pythagoras had uncovered for the first time the mathematical rule which governs a physical phenomenon and demonstrated that there was a fundamental relationship between mathematics and science. Ever since this discovery scientists have searched for the mathematical rules which appear to govern every single physical process and have found that numbers crop up in all manner of natural phenomena. For example, one particular number appears to guide the lengths of meandering rivers. Professor Hans-Henrik Stølum, an earth scientist at Cambridge University, has calculated the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies. Although the ratio varies from river to river, the average value is slightly greater than 3, that is to say that the actual length is roughly three times greater than the direct distance. In fact the ratio is approximately 3.14, which is close to the value of the number π, the ratio between the circumference of a circle and its diameter.

  The number π was originally derived from the geometry of circles and yet it reappears over and over again in a variety of scientific circumstances. In the case of the river ratio, the appearance of π is the result of a battle between order and chaos. Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which will in turn result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist, and so on. However, there is a natural process which will curtail the chaos: increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting. The river will become straighter and the loop will be left to one side forming an ox-bow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth. The ratio of π is most commonly found for rivers flowing across very gently sloping plains, such as those found in Brazil or the Siberian tundra.

  Pythagoras realised that numbers were hidden in everything, from the harmonies of music to the orbits of planets, and this led him to proclaim that ‘Everything is Number’. By exploring the meaning of mathematics, Pythagoras was developing the language which would enable him and others to describe the nature of the universe. Henceforth each breakthrough in mathematics would give scientists the vocabulary they needed to better explain the phenomena around them. In fact developments in mathematics would inspire revolutions in science.

  As well as discovering the law of gravity, Isaac Newton was a powerful mathematician. His greatest contribution to mathematics was his development of calculus, and in later years physicists would use the language of calculus to better describe the laws of gravity and to solve gravitational problems. Newton’s classical theory of gravity survived intact for centuries until it was superseded by Albert Einstein’s general theory of relativity, which developed a more detailed and alternative explanation of gravity. Einstein’s own ideas were only possible because of new mathematical concepts which provided him with a more sophisticated language for his more complex scientific ideas. Today the interpretation of gravity is once again being influenced by breakthroughs in mathematics. The very latest quantum theories of gravity are tied to the development of mathematical strings, a theory in which the geometrical and topological properties of tubes seem to best explain the forces of nature.

  Of all the links between numbers and nature studied by the Brotherhood, the most important was the relationship which bears their founder’s name. Pythagoras’ theorem provides us with an equation which is true of all right-angled triangles and which therefore also defines the right angle itself. In turn, the right angle defines the perpendicular, i.e. the relation of the vertical to the horizontal, and ultimately the relation between the three dimensions of our familiar universe. Mathematics, via the right angle, defines the very structure of the space in which we live.

  Figure 1. All right-angled triangles obey Pythagoras’ theorem.

  It is a profound realisation and yet the mathematics required to grasp Pythagoras’s theorem is relatively simple. To understand it, simply begin by measuring the length of the two short sides of a right-angled triangle (x and y), and then square each one (x2, y2). Then add the two squared numbers (x2 + y2) to give you a final number. If you work out this number for the triangle shown in Figure 1, then the answer is 25.

  You can now measure the longest side z, the so-called hypotenuse, and square this length. The remarkable result is that this number z2 is identical to the one you just calculated, i.e. 52 = 25. That is to say,

  In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

  Or in other words (or rather symbols):

  This is clearly true for the triangle in Figure 1, but what is remarkable is that Pythagoras’ theorem is true for every right-angled triangle you can possibly imagine. It is a universal law of mathematics, and you can rely on it whenever you come across any triangle with a right angle. Conversely if you have a triangle which obeys Pythagoras’ theorem, then you can be absolutely confident that it is a right-angled triangle.

  At this point it is important to note that, although this theorem will forever be associated with Pythagoras, it was actually used by the Chinese and the Babylonians one thousand years before. However, these cultures did not know that the theorem was true for every right-angled triangle. It was certainly true for the triangles they tested, but they had no way of showing that it was true for all the right-angled triangles which they had not tested. The reason for Pythagoras’ claim to the theorem is that it was he who first demonstrated its universal truth.

  But how did Pythagoras know that his theorem is true for every right-angled triangle? He could not hope to test the infinite variety of right-angled triangles, and yet he could still be one hundred per cent sure of the theorem’s absolute truth. The reason for his confidence lies in the concept of mathematical proof. The search for a mathematical proof is the search for a knowledge which is more absolute than the knowledge accumulated by any other discipline. The desire for ultimate truth via the method of proof is what has driven mathematicians for the last two and a half thousand years.

  Absolute Proof

  The story of Fermat’s Last Theorem revolves around the search for a missing proof. Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as u
nderstood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras.

  The idea of a classic mathematical proof is to begin with a series of axioms, statements which can be assumed to be true or which are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.

  Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value of such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favour of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.