Fermat's Last Theorem

by Simon Singh

  In the tenth century the French scholar Gerbert of Aurillac learnt the new counting system from the Moors of Spain and through his teaching positions at churches and schools throughout Europe he was able to introduce the new system to the West. In 999 he was elected Pope Sylvester II, an appointment which allowed him to further encourage the use of Indo-Arabic numerals. Although the efficiency of the system revolutionised accounting and was rapidly adopted by merchants, it did little to inspire a revival in European mathematics.

  The vital turning point for Western mathematics occurred in 1453 when the Turks ransacked Constantinople. During the intervening years the manuscripts which had survived the desecration of Alexandria had congregated in Constantinople, but once again they were threatened with destruction. Byzantine scholars fled westward with whatever texts they could preserve. Having survived the onslaught of Caesar, Bishop Theophilus, Caliph Omar and now the Turks, a few precious volumes of the Arithmetica made their way back to Europe. Diophantus was destined for the desk of Pierre de Fermat.

  Birth of a Riddle

  Fermat’s judicial responsibilities occupied a great deal of his time, but what little leisure he had was devoted entirely to mathematics. This was partly because judges in seventeenth-century France were discouraged from socialising on the grounds that friends and acquaintances might one day be called before the court. Fraternising with the locals would only lead to favouritism. Isolated from the rest of Toulouse’s high society, Fermat could concentrate on his hobby.

  There is no record of Fermat ever being inspired by a mathematical tutor; instead it was a copy of the Arithmetica which became his mentor. The Arithmetica sought to describe the theory of numbers, as it was in Diophantus’ time, via a series of problems and solutions. In effect Diophantus was presenting Fermat with one thousand years worth of mathematical understanding. In one book Fermat could find the entire knowledge of numbers as constructed by the likes of Pythagoras and Euclid. The theory of numbers had stood still ever since the barbaric burning of Alexandria, but now Fermat was ready to resume study of the most fundamental of mathematical disciplines.

  The Arithmetica which inspired Fermat was a Latin translation made by Claude Gaspar Bachet de Méziriac, reputedly the most learned man in all of France. As well as being a brilliant linguist, poet and classics scholar, Bachet had a passion for mathematical puzzles. His first publication was a compilation of puzzles entitled Problemes plaisans et délectables qui se font par les nombres, which included river-crossing problems, a liquid-pouring problem and several think-of-a-number tricks. One of the questions posed was a problem about weights:

  What is the least number of weights that can be used on a set of scales to weigh any whole number of kilograms from 1 to 40?

  Bachet had a cunning solution which shows that it is possible to achieve this task with only four weights. His solution is given in Appendix 4.

  Although he was merely a mathematical dilettante, Bachet’s interest in puzzles was enough for him to realise that Diophantus’ list of problems were on a higher plane and worthy of deeper study. He set himself the task of translating Diophantus’ opus and publishing it so that the techniques of the Greeks could be rekindled. It is important to realise that vast quantities of ancient mathematical knowledge had been completely forgotten. Higher mathematics was not taught in even the greatest European universities and it is only thanks to the efforts of scholars such as Bachet that so much was revived so rapidly. In 1621 when Bachet published the Latin version of the Arithmetica, he was contributing to the second golden age of mathematics.

  The Arithmetica contains over one hundred problems and for each one Diophantus gives a detailed solution. This level of conscientiousness was not a habit which Fermat ever picked up. Fermat was not interested in writing a textbook for future generations: he merely wanted to satisfy himself that he had solved a problem. While studying Diophantus’ problems and solutions, he would be inspired to think of and tackle other related and more subtle questions. Fermat would scribble down whatever was necessary to convince himself that he could see the solution and then he would not bother to write down the remainder of the proof. More often than not he would consign his inspirational jottings to the bin, and then move on to the next problem. Fortunately for us, Bachet’s publication of the Arithmetica contained generous margins on every page, and sometimes Fermat would hastily write logic and comments in these columns. These marginal notes would become an invaluable, if somewhat scanty, record of Fermat’s most brilliant calculations.

  One of Fermat’s discoveries concerned the so-called friendly numbers, or amicable numbers, closely related to the perfect numbers which had fascinated Pythagoras two thousand years earlier. Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220.

  The pair 220 and 284 was said to be symbolic of friendship. Martin Gardner’s book Mathematical Magic Show tells of talismans sold in the Middle Ages which were inscribed with these numbers on the grounds that wearing the charms would promote love. An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac. Early theologians noted that in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, was an expression of Jacob’s love for Esau.

  No other friendly numbers were identified until 1636 when Fermat discovered the pair 17,296 and 18,416. Although not a profound discovery, it demonstrates Fermat’s familiarity with numbers and his love of playing with them. Fermat started a fad for finding friendly numbers; Descartes discovered a third pair (9,363,584 and 9,437,056) and Leonhard Euler went on to list sixty-two amicable pairs. Curiously they had all overlooked a much smaller pair of friendly numbers. In 1866 a sixteen-year-old Italian, Nicolò Paganini, discovered the pair 1,184 and 1,210.

  During the twentieth century mathematicians have extended the idea further and have searched for so-called ‘sociable’ numbers, three or more numbers which form a closed loop. For example, in the loop of 5 numbers (12,496; 14,288; 15,472; 14,536; 14,264) the divisors of the first number add up to the second, the divisors of the second add to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.

  Although discovering a new pair of friendly numbers made Fermat something of a celebrity, his reputation was truly confirmed thanks to a series of mathematical challenges. For example, Fermat noticed that 26 is sandwiched between 25 and 27, one of which is a square number (25 = 52 = 5 × 5) and the other is a cube number (27 = 33 = 3 × 3 × 3). He searched for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. After days of strenuous effort he managed to construct an elaborate argument which proved without any doubt that 26 is indeed the only number between a square and a cube. His step-by-step logical proof established that no other numbers could fulfil this criterion.

  Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it? Despite the simplicity of the claim the proof is fiendishly complicated, and Fermat took particular delight in taunting the English mathematicians Wallis and Digby, who eventually had to admit defeat. Ultimately Fermat’s greatest claim to fame would turn out to be another challenge to the rest of the world. However, it would be an accidental riddle which was never intended for public discussion.

  The Marginal Note

  While studying Book II of the Arithmetica Fermat came upon
a whole series of observations, problems and solutions which concerned Pythagoras’ theorem and Pythagorean triples. For instance, Diophantus discussed the existence of particular triples which formed so-called ‘limping triangles’, ones in which the two shorter legs x and y differ only by one (e.g. x = 20, y = 21, z = 29 and 202 + 212 = 292).

  Fermat was struck by the variety and sheer quantity of Pythagorean triples. He was aware that centuries earlier Euclid had stated a proof, outlined in Appendix 5, which demonstrated that, in fact, there are an infinite number of Pythagorean triples. Fermat must have gazed at Diophantus’ detailed exposition of Pythagorean triples and wondered what there was to add to the subject. As he stared at the page he began to play with Pythagoras’ equation, trying to discover something which had evaded the Greeks. Suddenly, in a moment of genius which would immortalise the Prince of Amateurs, he created an equation which, though very similar to Pythagoras’ equation, had no solutions at all. This was the equation which the ten-year-old Andrew Wiles read about in the Milton Road Library.

  Instead of considering the equation

  Fermat was contemplating a variant of Pythagoras’ creation:

  As mentioned in the last chapter, Fermat had merely changed the power from 2 to 3, the square to a cube, but his new equation apparently had no whole number solutions whatsoever. Trial and error soon shows the difficulty of finding two cubed numbers which add together to make another cubed number. Could it really be the case that this minor modification turns Pythagoras’ equation, one with an infinite number of solutions, into an equation with no solutions?

  He altered the equation further by changing the power to numbers bigger than 3, and discovered that finding a solution to each of these equations was equally difficult. According to Fermat there appeared to be no three numbers which would perfectly fit the equation

  In the margin of his Arithmetica, next to Problem 8, he made a note of his observation:

  Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.

  It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.

  Among all the possible numbers there seemed to be no reason why at least one set of solutions could not be found, yet Fermat stated that nowhere in the infinite universe of numbers was there a ‘Fermatean triple’. It was an extraordinary claim, but one which Fermat believed he could prove. After the first marginal note outlining the theory, the mischievous genius jotted down an additional comment which would haunt generations of mathematicians:

  Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

  I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.

  This was Fermat at his most infuriating. His own words suggest that he was particularly pleased with this ‘truly marvellous’ proof, but he had no intention of bothering to write out the detail of the argument, never mind publishing it. He never told anyone about his proof, and yet despite his combination of indolence and modesty Fermat’s Last Theorem, as it would later be called, would become famous around the world for centuries to come.

  The Last Theorem Published at Last

  Fermat’s notorious discovery happened early in his mathematical career, in around 1637. Some thirty years later, while carrying out his judicial duties in the town of Castres, Fermat was taken seriously ill. On 9 January 1665, he signed his last arrêt, and three days later he died. Still isolated from the Parisian school of mathematics and not necessarily fondly remembered by his frustrated correspondents, Fermat’s discoveries were at risk of being lost forever. Fortunately Fermat’s eldest son, Clément-Samuel, who appreciated the significance of his father’s hobby, was determined that his discoveries should not be lost to the world. It is thanks to his efforts that we know anything at all about Fermat’s remarkable breakthroughs in number theory and, in particular, if it were not for Clément-Samuel, the enigma known as Fermat’s Last Theorem would have died with its creator.

  Clément-Samuel spent five years collecting his father’s notes and letters, and examining the jottings in the margins of his copy of the Arithmetica. The marginal note referring to Fermat’s Last Theorem was just one of many inspirational thoughts scribbled in the book, and Clément-Samuel undertook to publish these annotations in a special edition of the Arithmetica. In 1670 at Toulouse he brought out Diophantus’ Arithmetica Containing Observations by P. de Fermat. Alongside Bachet’s original Greek and Latin translations were forty-eight observations made by Fermat, one of which was to become known as Fermat’s Last Theorem.

  Once Fermat’s Observations reached the wider community, it was clear that the letters he had sent to colleagues represented mere morsels from a treasure trove of discovery. His personal notes contained a whole series of theorems. Unfortunately these were accompanied either with no explanation at all or with only a slight hint of the underlying proof. There were just enough tantalising glimpses of logic to leave mathematicians in no doubt that Fermat had proofs, but filling in the details was left as a challenge for them to take up.

  Leonhard Euler, one of the greatest mathematicians of the eighteenth century, attempted to prove one of Fermat’s most elegant observations, a theorem concerning prime numbers. A prime number is one which has no divisors – no number will divide into it without a remainder, except for 1 and the number itself. For instance, 13 is a prime number, but 14 is not. Nothing will divide into 13, but 2 and 7 will divide into 14. All prime numbers (except 2) can be put into two categories; those which equal 4n + 1 and those which equal 4n – 1, where n equals some number. So 13 is in the former group (4 × 3 + 1), whereas 19 is in the latter group (4 × 5 – 1). Fermat’s prime theorem claimed that the first type of primes were always the sum of two squares (13 = 22 + 32), whereas the second type could never be written in this way (19 = ?2 + ?2). This property of primes is beautifully simple, but trying to prove that it is true for every single prime number turns out to be remarkably difficult. For Fermat it was just one of many private proofs. The challenge for Euler was to rediscover Fermat’s proof. Eventually in 1749, after seven years work and almost a century after Fermat’s death, Euler succeeded in proving this prime number theorem.

  Fermat’s panoply of theorems ranged from the fundamental to the simply amusing. Mathematicians rank the importance of theorems according to their impact on the rest of mathematics. First, a theorem is considered important if it has a universal truth, that is to say, if it applies to an entire group of numbers. In the case of the prime number theorem, it is true not for just some prime numbers, but for all prime numbers. Second, theorems should reveal some deeper underlying truth about the relationship between numbers. A theorem can be the springboard for generating a whole host of other theorems, even inspiring the development of whole new branches of mathematics. Finally, a theorem is important if entire areas of research can be hindered for the lack of one logical link. Many mathematicians have cried themselves to sleep knowing that they could achieve a major result if only they could establish one missing link in their chain of logic.

  Because mathematicians employ theorems as stepping stones to other results, it was essential that every single one of Fermat’s theorems be proved. Just because Fermat said he had a proof of a theorem it could not be accepted at face value. Before it could be used, each theorem had to be proved with ruthless rigour, otherwise the consequences could have been disastrous. For example, imagine that mathematicians had accepted one of Fermat’s theorems. It would then be incorporated as a single element in a whole series of other larger proofs. In due course these larger proofs would be incorporated into even larger proofs, and so on. Ultimately hundreds of theorems could come to rely on the truth of the original unchecked theorem
. However, what if Fermat had made a mistake and the unchecked theorem was in fact flawed? All these other theorems which incorporated it would also be flawed, and vast areas of mathematics would collapse. Theorems are the foundations of mathematics, because once their truth has been established other theorems can safely be built on top of them. Unsubstantiated ideas are infinitely less valuable and are referred to as conjectures. Any logic which relies on a conjecture is itself a conjecture.

  Fermat said he had a proof for every one of his observations, so for him they were theorems. However, until the community at large could rediscover the individual proofs each one could only be considered a conjecture. In fact for the last 350 years Fermat’s Last Theorem should more accurately have been referred to as Fermat’s Last Conjecture.

  As the centuries passed, all his other observations were proved one by one, but Fermat’s Last Theorem stubbornly refused to give in so easily. In fact, it is called the ‘Last’ Theorem because it remains the last one of the observations to be proved. Three centuries of effort failed to find a proof, and this led to its notoriety as the most demanding riddle in mathematics. However, this acknowledged difficulty does not necessarily mean that Fermat’s Last Theorem is an important theorem in the ways described earlier. The Last Theorem, at least until very recently, seemed to fail to fulfil several criteria – it seemed that proving it would not lead to anything profound, it would not give any particularly deep insight about numbers, and it would not help prove any other conjectures.