First, let us look at connecting the vertex to itself with this additional line. As shown in Figure 8, when the line is added, this also results in a new region. Therefore the network formula remains true because the extra region (+1) cancels the extra line (–1). If further lines are added in this way the network formula will still remain true because each new line will create a new region.
Figure 8. Euler proved his network formula by showing that it was true for the simplest network, and then demonstrating that the formula would remain true whatever extensions were added to the single vertex.
Second, let us look at using the line to connect the original vertex to a new vertex, as shown in Figure 8. Once again the network formula remains true because the extra vertex (+1) cancels the extra line (–1). If further lines are added in this way, the network formula will still remain true because each new line will create a new vertex.
This was all that Euler required for his proof. He argued that the network formula was true for the simplest of all networks, the single vertex. Furthermore, all other networks, no matter how complicated, can be constructed from the simplest network by adding lines one at a time. Each time a new line is added the network formula will remain true because either a new vertex or a new region will always be added and this will have a compensating effect. Euler had developed a simple but powerful strategy. He proved that the formula is true for the most basic network, a single vertex, and then he demonstrated that any operation which complicated the network would continue to conserve the validity of the formula. Therefore the formula is true for the infinity of all possible networks.
When Euler first encountered Fermat’s Last Theorem, he must have hoped that he could solve it by adopting a similar strategy. The Last Theorem and the network formula come from very different areas of mathematics but they have one thing in common, which is that both say something about an infinite number of objects. The network formula says that for the infinite number of networks that exist the number of vertices and regions less the number of lines always equals 1. Fermat’s Last Theorem claims that for an infinite number of equations there are no whole number solutions. Recall that Fermat stated that there are no whole number solutions to the following equation:
This equation represents an infinite set of equations:
Euler wondered if he could prove that one of the equations had no solutions and then extrapolate the result to all the remaining equations, in the same way he had proved his network formula for all networks by generalising it from the simplest case, the single vertex.
Euler’s task was given a head start when he discovered a clue hidden in Fermat’s jottings. Although Fermat never wrote down a proof for the Last Theorem, he did cryptically describe a proof for the specific case n = 4 elsewhere in his copy of the Arithmetica and incorporated it into the proof of a completely different problem. Even though this is the most complete calculation he ever committed to paper, the details are still sketchy and vague, and Fermat concludes the proof by saying that lack of time and paper prevent him from giving a fuller explanation. Despite the lack of detail in Fermat’s scribbles, they clearly illustrate a particular form of proof by contradiction known as the method of infinite descent.
In order to prove that there were no solutions to the equation x4 + y4 = z4, Fermat began by assuming that there was a hypothetical solution
By examining the properties of (X1, r1, Z1), Fermat could demonstrate that if this hypothetical solution did exist then there would have to be a smaller solution (X2, r2, Z2). Then, by examining this new solution, Fermat could show there would be an even smaller solution (X3, r3, Z3), and so on.
Fermat had discovered a descending staircase of solutions, which theoretically would continue forever, generating ever-smaller numbers. However, x, y and z must be whole numbers, and so the never-ending staircase is impossible because there must be a smallest possible solution. This contradiction proves that the initial assumption that there is a solution (X1, r1, Z1) must be false. Using the method of infinite descent Fermat had demonstrated that it is forbidden for the equation with n = 4 to have any solutions, because otherwise the consequences would be absurd.
Euler tried to use this as a starting point for constructing a general proof for all the other equations. As well as building up to n = infinity, he would also have to build down to n = 3 and it was this single downward step which he attempted first. On 4 August 1753 Euler announced in a letter to the Prussian mathematician Christian Goldbach that he had adapted Fermat’s method of infinite descent and successfully proved the case for n = 3. After a hundred years this was the first time anybody had succeeded in making any progress towards meeting Fermat’s challenge.
In order to extend Fermat’s proof from n = 4 to cover the case n = 3 Euler had to incorporate the bizarre concept of a so-called imaginary number, an entity which had been discovered by European mathematicians in the sixteenth century. It is strange to think of new numbers being ‘discovered’, but this is mainly because we are so familiar with the numbers we commonly use that we forget that there was a time when some of these numbers were not known. Negative numbers, fractions and irrational numbers all had to be discovered and the motivation in each case was to answer otherwise unanswerable questions.
The history of numbers begins with the simple counting numbers (1,2,3, …) otherwise known as natural numbers. These numbers are perfectly satisfactory for adding together simple whole quantities, such as sheep or gold coins, to achieve a total number which is also a whole quantity. As well as addition, the other simple operation of multiplication also acts upon whole numbers to generate other whole numbers. However, the operation of division throws up an awkward problem. While 8 divided by 2 equals 4, we find that 2 divided by 8 equals ¼. The result of the latter division is not a whole number but a fraction.
Division is a simple operation performed on natural numbers which requires us to look beyond the natural numbers in order to obtain the answer. It is unthinkable for mathematicians not, in theory at least, to be able to answer every single question, and this necessity is called completeness. There are certain questions concerning natural numbers which would be unanswerable without resorting to fractions. Mathematicians express this by saying that fractions are necessary for completeness.
It is this need for completeness which led the Hindus to discover negative numbers. The Hindus noticed that, while 3 subtracted from 5 was obviously 2, subtracting 5 from 3 was not such a simple matter. The answer was beyond the natural counting numbers, and could only be accommodated by introducing the concept of negative numbers. Some mathematicians did not accept this extension into abstraction and referred to negative numbers as ‘absurd’ or ‘fictitious’. While an accountant could hold one gold coin, or even half a gold coin, it was impossible to hold a negative coin.
The Greeks also had a yearning for completeness and this led them to discover irrational numbers. In Chapter 2 the question arose, What number is the square root of two, √2? The Greeks knew that this number was roughly equal to 7⁄7, but when they tried to discover the exact fraction they found that it did not exist. Here was a number which could never be represented as a fraction, but this new type of number was necessary in order to answer a simple question, What is the square root of two? The demand for completeness meant that yet another colony was added to the empire of numbers.
Figure 9. All numbers can be positioned along the number line, which extends to infinity in both directions.
By the Renaissance, mathematicians assumed that they had discovered all the numbers in the universe. All numbers could be thought of as lying on a number line, an infinitely long line with zero at the centre, as shown in Figure 9. The whole numbers were spaced equally along the number line, with the positive numbers on the right of zero extending to positive infinity and the negative numbers on the left of zero extending to negative infinity. The fractions occupied the spaces between the whole numbers, and the irrational numbers were interspersed between the frac
tions.
The number line suggested that completeness had apparently been achieved. All the numbers seemed to be in place, ready to answer all mathematical questions – in any case, there was no more room on the number line for any new numbers. Then during the sixteenth century there were renewed rumblings of disquiet. The Italian mathematician Rafaello Bombelli was studying the square roots of various numbers when he stumbled upon an unanswerable question.
The problem began by asking, What is the square root of one, √1? The obvious answer is 1, because 1 × 1 = 1. The less obvious answer is –1. A negative number multiplied by another negative number generates a positive number. This means –1 × –1 = +1. So, the square root of +1 is both +1 and –1. This abundance of answers is fine, but then the question arises, What is the square root of negative one, √–1? The problem seems to be intractable. The solution cannot be +1 or –1, because the square of both these numbers is +1. However, there are no other obvious candidates. At the same time completeness demands that we must be able to answer the question.
The solution for Bombelli was to create a new number, i, called an imaginary number, which was simply defined as the solution to the question, What is the square root of negative one? This might seem like a cowardly solution to the problem, but it was no different to the way in which negative numbers were introduced. Faced with an otherwise unanswerable question the Hindus merely defined –1 as the solution to the question, What is zero subtract one? It is easier to accept the concept of –1 only because we have experience of the analogous concept of ‘debt’, whereas we have nothing in the real world to underpin the concept of an imaginary number. The seventeenth-century German mathematician Gottfried Leibniz elegantly described the strange nature of the imaginary number: ‘The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being.’
Once we have defined i as being the square root of –1, then 2i must exist, because this would be the sum of i plus i (as well as being the square root of –4). Similarly i⁄2 must exist because this is the result of dividing i by 2. By performing simple operations it is possible to achieve an imaginary equivalent of every so-called real number. There are imaginary counting numbers, imaginary negative numbers, imaginary fractions and imaginary irrationals.
Figure 10. The introduction of an axis for imaginary numbers turns the number line into a number plane. Any combination of real and imaginary numbers has a position on the number plane.
The problem which now arises is that all these imaginary numbers have no natural position along the real number line. Mathematicians resolve this crisis by creating a separate imaginary number line which is perpendicular to the real one, and which crosses at zero, as shown in Figure 10. Numbers are now no longer restricted to a one-dimensional line, but rather they occupy a two-dimensional plane. While pure imaginary or pure real numbers are restricted to their respective lines, combinations of real and imaginary numbers (e.g. 1 + 2i), called complex numbers, live on the so-called number plane.
What is particularly remarkable is that complex numbers can be used to solve any conceivable equation. For example, in order to calculate √(3 + 4i), mathematicians do not have to resort to inventing a new type of number – the answer turns out to be 2 + i, another complex number. In other words the imaginary numbers appear to be the final element required to make mathematics complete.
Although the square roots of negative numbers have been referred to as imaginary numbers, mathematicians consider i no more abstract than a negative number or any counting number. In addition, physicists discovered that imaginary numbers provide the best language for describing some real-world phenomena. With a few minor manipulations imaginary numbers turn out to be the ideal way to analyse the natural swinging motion of objects such as pendula. This motion, technically called a sinusoidal oscillation, is found throughout nature, and so imaginary numbers have become an integral part of many physical calculations. Nowadays electrical engineers conjure up i to analyse oscillating currents, and theoretical physicists calculate the consequences of oscillating quantum mechanical wave functions by summoning up the powers of imaginary numbers.
Pure mathematicians have also exploited imaginary numbers, using them to find answers to previously impenetrable problems. Imaginary numbers literally add a new dimension to mathematics, and Euler hoped to exploit this extra degree of freedom to attack Fermat’s Last Theorem.
In the past other mathematicians had tried to adapt Fermat’s method of infinite descent to work for cases other than n = 4, but in every case attempts to stretch the proof only led to gaps in the logic. However, Euler showed that by incorporating the imaginary number, i, into his proof he could plug holes in the proof, and force the method of infinite descent to work for the case n = 3.
It was a tremendous achievement, but one which he could not repeat for other cases of Fermat’s Last Theorem. Unfortunately Euler’s endeavours to make the argument work for the cases up to infinity all ended in failure. The man who created more mathematics than anybody else in history was humbled by Fermat’s challenge. His only consolation was that he had made the first breakthrough in the world’s hardest problem.
Undaunted by this failure Euler continued to create brilliant mathematics until the day he died, an achievement made all the more remarkable by the fact that during the final years of his career he was totally blind. His loss of sight began in 1735 when the Academy in Paris offered a prize for the solution to an astronomical problem. The problem was so awkward that the mathematical community asked the Academy to allow them several months in which to come up with an answer, but for Euler this was unnecessary. He became obsessed with the task, worked continually for three days and duly won the prize. However, poor working conditions combined with intense stress cost Euler, then still only in his twenties, the sight of one eye. This is apparent in many portraits of Euler.
On the advice of Jean Le Rond d’Alembert, Euler was replaced by Joseph-Louis Lagrange as mathematician to the court of Frederick the Great, who later commented: ‘To your care and recommendation am I indebted for having replaced a half-blind mathematician with a mathematician with both eyes, which will especially please the anatomical members of my Academy.’ Euler returned to Russia where Catherine the Great welcomed back her ‘mathematical cyclops’.
The loss of one eye was only a minor handicap – in fact Euler claimed that ‘now I will have less distraction’. Forty years later, at the age of sixty, his situation worsened considerably, when a cataract in Euler’s good eye meant he was destined to become completely blind. He was determined not to give in and began to practise writing with his fading eye closed in order to perfect his technique before the onset of darkness. Within weeks he was blind. The rehearsal paid off for a while, but a few months later Euler’s script became illegible, whereupon his son Albert acted as his amanuensis.
Euler continued to produce mathematics for the next seventeen years and, if anything, he was more productive than ever. His immense intellect allowed him to juggle concepts without having to commit them to paper, and his phenomenal memory allowed him to use his own brain as a mental library. Colleagues suggested that the onset of blindness appeared to expand the horizons of his imagination. It is worth noting that Euler’s computations of lunar positions were completed during his period of blindness. For the emperors of Europe this was the most prized of mathematical achievements, a problem that had confounded the greatest mathematicians in Europe, including Newton.
In 1776 an operation was performed to remove the cataract, and for a few days Euler’s sight seemed to have been restored. Then infection set in and Euler was plunged back into darkness. Undaunted he continued to work until, on 18 September 1783, he suffered a fatal stroke. In the words of the mathematician-philosopher the Marquis de Condorcet, ‘Euler ceased to live and calculate.’
A Petty Pace
A century after Fermat’s death there existed proofs for only two specific cases of the
Last Theorem. Fermat had given mathematicians a head start by providing them with the proof that there were no solutions to the equation
Euler had adapted the proof to show that there were no solutions to
After Euler’s breakthrough it was still necessary to prove that there were no whole number solutions to an infinity of equations:
Although mathematicians were making embarrassingly slow progress, the situation was not quite as bad as it might seem at first sight. The proof for the case n = 4 also proves the cases n = 8, 12, 16, 20, …. The reason is that any number which can be written as an 8th (or a 12th, 16th, 20th, …) power can also be rewritten as a 4th power. For instance, the number 256 is equal to 28, but it is also equal to 44. Therefore any proof which works for the 4th power will also work for the 8th power and for any other power that is a multiple of 4. Using the same principle, Euler’s proof for the case n = 3 automatically proves the cases n = 6, 9, 12, 15, …
Suddenly, the numbers are tumbling and Fermat looks vulnerable. The proof for the case n = 3 is particularly significant because the number 3 is an example of a prime number. As explained earlier, a prime number has the special property of not being the multiple of any whole number except for 1 and itself. Other prime numbers are 5, 7, 11, 13, …. All the remaining numbers are multiples of the primes, and are referred to as non-primes, or composite numbers.
Number theorists consider prime numbers to be the most important numbers of all because they are the atoms of mathematics. Prime numbers are the numerical building blocks because all other numbers can be created by multiplying combinations of the prime numbers. This seems to lead to a remarkable breakthrough. To prove Fermat’s Last Theorem for all values of n, one merely has to prove it for the prime values of n. All other cases are merely multiples of the prime cases and would be proved implicitly.