Fermat's Last Theorem (25 page)

They got their name because in the past they were used to measure the perimeters of ellipses and the lengths of planetary orbits, but for clarity I will simply refer to them as
elliptic equations
rather than elliptic curves.

The challenge with elliptic equations, as with Fermat's Last Theorem, is to figure out if they have whole number solutions, and, if so, how many. For example, the elliptic equation

has only one set of whole number solutions, namely

Proving that this elliptic equation has only one set of whole number solutions is an immensely difficult task, and in fact it was Pierre de Fermat who discovered the proof. You might remember
that in
Chapter 2
it was Fermat who proved that 26 is the only number in the universe sandwiched between a square and a cube number. This is equivalent to showing that the above elliptic equation has only one solution, i.e. 5
²
and 3
³
are the only square and cube that differ by 2, and therefore 26 is the only number that can be sandwiched between two such numbers.

What makes elliptic equations particularly fascinating is that they occupy a curious niche between other simpler equations which are almost trivial and other more complicated equations which are impossible to solve. By simply changing the values of
a, b
and
c
in the general elliptic equation mathematicians can generate an infinite variety of equations, each one with its own characteristics, but all of them just within the realm of solubility.

Elliptic equations were originally studied by the ancient Greek mathematicians, including Diophantus who devoted large parts of his
Arithmetica
to exploring their properties. Probably inspired by Diophantus, Fermat also took up the challenge of elliptic equations, and, because they had been studied by his hero, Wiles was happy to explore them further. Even after two thousand years elliptic equations still offered formidable problems for students such as Wiles: ‘They are very far from being completely understood. There are many apparently simple questions I could pose on elliptic equations that are still unresolved. Even questions that Fermat himself considered are still unresolved. In some way all the mathematics that I've done can trace its ancestry to Fermat, if not Fermat's Last Theorem.'

In the equations which Wiles studied as a graduate student, determining the exact number of solutions was so difficult that the only way to make any progress was to simplify the problem. For example, the following elliptic equation is almost impossible to tackle directly:

The challenge is to figure out how many whole number solutions there are to the equation. One fairly trivial solution is
x
= 0 and
y
= 0:

A slightly more interesting solution is
x
= 1 and
y
= 0:

There may be other solutions but, with an infinite quantity of whole numbers to investigate, giving a complete list of solutions to this particular equation is an impossible task. A simpler task is to look for solutions within a finite number space, so-called clock arithmetic.

Earlier we saw how numbers can be thought of as marks along the number line which extends to infinity, as shown in
Figure 13
. To make the number space finite, clock arithmetic involves truncating the line and looping it back on itself to form a number ring as opposed to a number line.
Figure 14
shows a 5-clock, where the number line has been truncated at 5 and looped back to 0. The number 5 vanishes and becomes equivalent to 0, and therefore the only numbers in 5-clock arithmetic are 0, 1, 2, 3, 4.

Figure 13. Conventional arithmetic can be thought of as movements up and down the number line.

In normal arithmetic we can think of addition as moving along the line a certain number of spaces. For example, 4 + 2 = 6 is the same as saying: begin at 4, and move along the number line 2 spaces, and arrive at 6.

However, in 5-clock arithmetic:

This is because if we start at 4 and move round 2 spaces then we arrive back at 1. Clock arithmetic might appear unfamiliar but in fact, as the name suggests, it is used every day when people discuss the time. Four hours after 11 o'clock (that is to say, 11 = 4) is generally not called 15 o'clock, but rather 3 o'clock. This is 12-clock arithmetic.

As well as addition we can perform all the other common mathematical operations, such as multiplication. In 12-clock arithmetic 5 × 7 = 11. This multiplication can be thought of as follows: if you start at 0, then move along 5 lots of 7 spaces, you will eventually arrive at 11. Although this is one way of thinking about multiplication in clock arithmetic, there are short cuts which speed up calculations. For example, to calculate 5 × 7 in 12-clock arithmetic, we can begin by just working out the normal result which is 35. We then divide 35 by 12 and work out the remainder, which is the answer to the original question. So 12 goes into 35 only twice, with a remainder of 11, and sure enough 5 × 7 in 12-clock arithmetic is 11. This is equivalent to imagining going around the clock twice and still having 11 spaces to travel.

Because clock arithmetics only deal with a limited number space, it is relatively easy to work out all the possible solutions to an elliptic equation for a given clock arithmetic. For example, working in 5-clock arithmetic it is possible to list all the possible solutions to the elliptic equation