Fermat's Last Theorem (43 page)

Then, by Axiom 2,

Bearing in mind that
k
+
l
= 0, we know that

By applying Axiom 4, we can at last declare what we set out to prove:

Let us examine Mr Black's options. First, Mr Black could aim at Mr Grey. If he is successful then the next shot will be taken by Mr White. Mr White has only one opponent left, Mr Black, and as Mr White is a perfect shot then Mr Black is a dead man.

A better option is for Mr Black to aim at Mr White. If he is successful then the next shot will be taken by Mr Grey. Mr Grey hits his target only two times out of three and so there is a chance that Mr Black will survive to fire back at Mr Grey and possibly win the truel.

It appears that the second option is the strategy which Mr Black should adopt. However, there is a third and even better option. Mr Black could aim into the air. Mr Grey has the next shot and he will aim at Mr White, because he is the more dangerous opponent. If Mr White survives then he will aim at Mr Grey because he is the more dangerous opponent. By aiming into the air, Mr Black is allowing Mr Grey to eliminate Mr White or vice versa.

This is Mr Black's best strategy. Eventually Mr Grey or Mr White will die and then Mr Black will aim at whoever survives. Mr Black has manipulated the situation so that, instead of having the first shot in a truel, he has first shot in a duel.

Mathematicians find it useful to have neat formulae which give the sum of various lists of numbers. In this case the challenge is to find a formula which gives the sum of the first
n
counting numbers.

For example, the sum of just the first number is 1, the sum of the first two numbers is 3 (i.e. 1 + 2), the sum of the first three numbers is 6 (i.e. 1 + 2 + 3), the sum of the first four numbers is 10 (i.e. 1 + 2 + 3 + 4), and so on.

A candidate formula which seems to describe this pattern is:

In other words if we want to find the sum of the first
n
numbers, then we simply enter that number into the formula above and work out the answer.

Proof by induction can prove that this formula works for every number up to infinity.

The first step is to show that the formula works for the first case,
n
= 1. This is fairly straightforward, because we know that the sum of just the first number is 1, and if we enter
n
= 1 into the candidate formula we get the correct result:

The first domino has been toppled.

The next step in proof by induction is to show that if the formula is true for any value
n
, then it must also be true for
n
+ 1. If

then,

After rearranging and regrouping the terms on the right, we get

What is important to note here is that the form of this new equation is exactly the same as the original equation except that every appearance of
n
has been replaced by (
n
+ 1).

In other words, if the formula is true for
n
, then it must also be true for
n
+ 1. If one domino falls, it will always knock over the next one. The proof by induction is complete.

In researching this book I have relied on numerous books and articles. In addition to my main sources for each chapter, I have also listed other material which may be of interest to both the general reader and experts in the field. Where the tide of the source does not indicate its relevance I have given a sentence or two describing its contents.

The Last Problem
, by E.T. Bell, 1990, Mathematical Association of America. A popular account of the origins of Fermat's Last Theorem.

Pythagoras – A Short Account of His Life and Philosophy
, by Leslie Ralph, 1961, Krikos.

Pythagoras – A Life
, by Peter Gorman, 1979, Routledge and Kegan Paul.

A History of Greek Mathematics
, Vols. 1 and 2, by Sir Thomas Heath, 1981, Dover.

Mathematical Magic Show
, by Martin Gardner, 1977, Knopf. A collection of mathematical puzzles and riddles.

River meandering as a self-organization process, by Hans-Henrik Støllum,
Science
271
(1996), 1710-1713.

The Mathematical Career of Pierre de Fermat
, by Michael Mahoney, 1994, Princeton University Press. A detailed investigation into the life and work of Pierre de Fermat.

Archimedes' Revenge
, by Paul Hoffman, 1988, Penguin. Fascinating tales which describe the joys and perils of mathematics.

Men of Mathematics
, by E.T. Bell, Simon and Schuster, 1937. Biographies of history's greatest mathematicians, including Euler, Fermat, Gauss, Cauchy and Kummer.

The periodical cicada problem, by Monte Lloyd and Henry S. Dybas,
Evolution
20
(1966), 466-505.

Women in Mathematics
, by Lynn M. Osen, 1994, MIT Press. A largely non-mathematical text containing the biographies of many of the foremost female mathematicians in history, including Sophie Germain.

Math Equals: Biographies of Women Mathematicians+Related Activities
, by Teri Perl, 1978, Addison-Wesley.

Women in Science
, by H.J. Mozans, 1913, D. Appleton and Co.

Sophie Germain, by Amy Dahan Dalmédico,
Scientific American
, December 1991. A short article describing the life and work of Sophie Germain.

Fermat's Last Theorem – A Genetic Introduction to Algebraic Number Theory
, by Harold M. Edwards, 1977, Springer. A mathematical discussion of Fermat's Last Theorem, including detailed outlines of some of the early attempts at a proof.

Elementary Number Theory
, by David Burton, 1980, Allyn & Bacon.

Various communications, by A. Cauchy,
C. R. Acad. Sci. Paris
24
(1847), 407–416, 469–483.

Note au sujet de la demonstration du theoreme de Fermat, by G. Lamé,
C. R. Acad. Sci. Paris
24
(1847), 352.