Fermat's Last Theorem (15 page)

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 2
⁸
, but it is also equal to 4
⁴
. 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.

Intuitively this enormously simplifies the problem, because you can ignore those equations which involve a value of
n
that is not a prime number. The number of equations remaining is now vastly reduced. For example, for the values of
n
up to 20, there are only six values which need to be proved:

If one can prove Fermat's Last Theorem for just the prime values of
n
, then the theorem is proved for all values of
n.
If one considers all whole numbers, then it is obvious that there are infinitely many. If one considers just the prime numbers, which are only a small fraction of all the whole numbers, then surely the problem is much simpler?

Intuition would suggest that if you begin with an infinite quantity and then remove the bulk of it, then you would expect to be left with something finite. Unfortunately intuition is not the arbiter of truth in mathematics, but rather logic. In fact, it is possible to prove that the list of primes is never-ending. Therefore, despite being able to ignore the vast majority of equations relating to non-prime values of
n
, the remaining equations relating to prime values of
n
are still infinite in number.

The proof that there is an infinity of primes dates all the way back to Euclid, and is one of the classic arguments of mathematics. Initially Euclid assumes that there is a finite list of known prime numbers, and then shows that there must exist an infinite number of additions to this list. There are
N
prime numbers in Euclid's finite list, which are labelled
P
₁
,
P
₂
,
P
₃
, …,
P
_N
.
Euclid can then generate a new number Q
_A
such that

This new number
Q
_A
is either prime or not prime. If it is prime then we have succeeded in generating a new, bigger prime number, and therefore our original list of primes was not complete. On the other hand, if
Q
_A
is not prime, then it must be perfectly divisible by a prime. This prime cannot be one of the known primes because dividing
Q
_A
by any of the known primes will inevitably lead to a remainder of 1. Therefore there must be some new prime, which we can call
P
_N
+ 1.

We have now arrived at the stage where either
Q
_A
is a new prime or we have another new prime
P
_N+1
. Either way we have added to our original list of primes. We can now repeat the process, including our new prime (
P
_N+1
or
Q
_A
) in our list, and generate some new number
Q
_B
. Either this new number will be yet another new prime, or there will have to be some other new prime
P
_N+2
that is not on our list of known primes. The upshot of the argument is that, however long our list of prime numbers, it is always possible to find a new one. Therefore the list of primes is never-ending and infinite.

But how can something which is undeniably smaller than an infinite quantity also be infinite? The German mathematician David Hilbert once said: ‘The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.' To resolve the paradox of the infinite it is necessary to define what is meant by infinity. Georg Cantor, who worked alongside Hilbert, defined infinity as the size of the never-ending list of counting numbers (1, 2, 3, 4, …). Consequently anything which is comparable in size is equally infinite.

By this definition the number of even counting numbers, which would intuitively appear to be smaller, is also infinite. It is easy to
demonstrate that the quantity of counting numbers and the quantity of even numbers are comparable because we can pair off each counting number with a corresponding even number:

If every member of the counting numbers list can be matched up with a member of the even numbers list then the two lists must be the same size. This method of comparison leads to some surprising conclusions, including the fact that there are an infinite number of primes. Although Cantor was the first person to tackle infinity in a formal way, he was initially heavily criticised by the mathematical community for his radical definition. Towards the end of his career the attacks became increasingly personal and this resulted in Cantor suffering mental illness and severe depression. Eventually, after his death, his ideas became widely accepted as the only consistent, accurate and powerful definition of infinity. As a tribute Hilbert said: ‘No one shall drive us from the paradise Cantor has created for us.'

Hilbert went on to create an example of infinity, known as
Hilbert's Hotel
, which clearly illustrates its strange qualities. This hypothetical hotel has the desirable attribute of having an infinite number of rooms. One day a new guest arrives and is disappointed to learn that, despite the hotel's infinite size, all the rooms are occupied. Hilbert, the clerk, thinks for a while and then reassures the new arrival that he will find an empty room. He asks all his current guests to move to the next room, so that the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on. Everybody who was in the hotel still has a room, which allows the
new arrival to slip into the vacant room 1. This shows that infinity plus one equals infinity.

The following night Hilbert has to deal with a much greater problem. The hotel is still full when an infinitely large coach arrives with an infinite number of new guests. Hilbert remains unperturbed and rubs his hands at the thought of infinitely more hotel bills. He asks all his current guests to move to the room which is double the number of their current room. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. Everybody who was in the hotel still has a room and yet an infinite number of rooms, all the odd ones, have been vacated for the new arrivals. This shows that double infinity is still infinity.

Hilbert's Hotel seems to suggest that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel – the infinity of even numbers can be matched up and compared with the infinity of all counting numbers. However, some infinities are indeed bigger than others. For example, any attempt to pair every rational number with every irrational number ends in failure, and in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers. Mathematicians have had to develop a whole system of nomenclature to deal with the varying scales of infinity and conjuring with these concepts is one of today's hottest topics.

Although the infinity of primes dashed hopes for an early proof of Fermat's Last Theorem, a countless supply of prime numbers does have more positive implications in other areas such as espionage and the evolution of insects. Before returning to the quest for a proof of Fermat's Last Theorem it is worth briefly investigating the uses and abuses of primes.