Fermat's Last Theorem (13 page)

where

V
= the number of vertices (intersections) in the network,

L
= the number of lines in the network,

R
= the number of regions (enclosed areas) in the network.

Euler claimed that for any network one could add the number of vertices and regions and subtract the number of lines and the total would always be 1. For example, all the networks in
Figure 7
obey the rule.

Figure 7. All conceivable networks obey Euler's network formula.

It is possible to imagine testing this formula on a whole series of networks and if it turned out to be true on each occasion it would be tempting to assume that the formula is true for all networks. Although this might be enough evidence for a scientific theory, it is inadequate to justify a mathematical theorem. The only way to show that the formula works for every possible network is to construct a foolproof argument, which is exactly what Euler did.

Euler began by considering the simplest network of all, i.e. a single vertex as shown in
Figure 8
. For this network the formula is clearly true: there is one vertex, and no lines or regions, and therefore

Euler then considered what would happen if he added something to this simplest of all networks. Any extension to the single vertex requires the addition of a line. The line can either connect the existing vertex to itself, or it can connect the existing vertex to a new vertex.

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
x
⁴
+
y
⁴
=
z
⁴
, Fermat began by assuming that there was a hypothetical solution

By examining the properties of (
X
₁
,
r
₁
,
Z
₁
), Fermat could
demonstrate that if this hypothetical solution did exist then there would have to be a smaller solution (
X
₂
,
r
₂
,
Z
₂
). Then, by examining this new solution, Fermat could show there would be an even smaller solution (
X
₃
,
r
₃
,
Z
₃
), 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 (
X
₁
,
r
₁
,
Z
₁
) 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.