Fermat's Last Theorem (10 page)

This may seem a trivial step forward to the modern reader, but the deeper meaning of the zero symbol had been ignored by all the ancient Greek philosophers, including Aristotle. He had argued that the number zero should be outlawed because it disrupted the consistency of the other numbers – dividing any ordinary number by zero led to an incomprehensible result. By the sixth century the Indian mathematicians no longer brushed this problem under the rug, and the seventh-century scholar Brahmagupta was sophisticated enough to use division by zero as a definition for infinity.

While Europe had abandoned the noble search for truth, India and Arabia were consolidating the knowledge which had been smuggled out of the embers of Alexandria and were reinterpreting it in a new and more eloquent language. As well as adding zero to the mathematical vocabulary, they replaced the primitive Greek symbols and cumbersome Roman numerals with the counting system which has now been universally adopted. Once again, this might seem like an absurdly humble step forward, but try multiplying CLV by DCI and you will appreciate the significance of the breakthrough. The equivalent task of multiplying 155 by 601 is a good deal simpler. The growth of any discipline depends on the ability to communicate and develop ideas, and this in turn relies on a language which is sufficiently detailed and flexible. The ideas of Pythagoras and Euclid were no less elegant for their awkward expression, but translated into the symbols of Arabia they would blossom and give fruit to newer and richer concepts.

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.

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 = 5
²
= 5 × 5) and the other is a cube number (27 = 3
³
= 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.

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 20
²
+ 21
²
= 29
²
).

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