Fermat's Last Theorem (19 page)

One example of a problem which evaded solution for decades is the
dot conjecture.
The challenge involves a series of dots which are all connected to each other by straight lines, such as the dot diagrams shown in
Figure 11
. The conjecture claims that it is impossible to draw a dot diagram such that every line has at least three dots on it (excluding the diagram where all the dots are on the same line). Certainly by experimenting with a few diagrams this appears to be true. For example,
Figure 11(a)
has five dots connected by six lines. Four of the lines do not have three dots on them and so clearly this arrangement does not satisfy the requirement that all lines have three dots. By adding an extra dot and the associated line, as in
Figure 11(b)
, the number of lines which do not have three dots is reduced to just three. However, trying to adapt the diagram further so that all the lines have three dots appears to be impossible. Of course, this does not prove that no such diagram exists.

Generations of mathematicians tried and failed to find a proof of the apparently straightforward dot conjecture. What made the
conjecture even more infuriating was that when a proof was eventually discovered, it involved only a minimal amount of mathematical knowledge mixed with a little extra cunning. The proof is outlined in
Appendix 6
.

There was a possibility that all the techniques required to prove Fermat's Last Theorem were available, and that the only missing ingredient was ingenuity. Wiles was not prepared to give up: finding a proof of the Last Theorem had turned from being a childhood fascination in to a fully fledged obsession. Having learnt all there was to learn about the mathematics of the nineteenth century, Wiles decided to arm himself with techniques of the twentieth century.

Proof is an idol before which the mathematician tortures himself.

Sir Arthur Eddington

Following the work of Ernst Kummer, hopes of finding a proof for the Last Theorem seemed fainter than ever. Furthermore mathematics was beginning to move into different areas of study and there was a risk that the new generation of mathematicians would ignore what seemed an impossible dead-end problem. By the beginning of the twentieth century the problem still held a special place in the hearts of number theorists, but they treated Fermat's Last Theorem in the same way that chemists treated alchemy. Both were foolish romantic dreams from a past age.

Then in 1908 Paul Wolfskehl, a German industrialist from Darmstadt, gave the problem a new lease of life. The Wolfskehl family were famous for their wealth and their patronage of the arts and sciences, and Paul was no exception. He had studied mathematics at university and, although he devoted most of his life to building the family's business empire, he maintained contact with professional mathematicians and continued to dabble in number theory. In particular Wolfskehl refused to give up on Fermat's Last Theorem.

Wolfskehl was by no means a gifted mathematician and he was
not destined to make a major contribution to finding a proof of the Last Theorem. Nonetheless, thanks to a curious chain of events, he was to become forever associated with Fermat's problem, and would inspire thousands of others to take up the challenge.

The story begins with Wolfskehl's obsession with a beautiful woman, whose identity has never been established. Depressingly for Wolfskehl the mysterious woman rejected him and he was left in such a state of utter despair that he decided to commit suicide. He was a passionate man, but not impetuous, and he planned his death with meticulous detail. He set a date for his suicide and would shoot himself through the head at the stroke of midnight. In the days that remained he settled all his outstanding business affairs, and on the final day he wrote his will and composed letters to all his close friends and family.

Wolfskehl had been so efficient that everything was completed slightly ahead of his midnight deadline, so to while away the hours he went to the library and began browsing through the mathematical publications. It was not long before he found himself staring at Kummer's classic paper explaining the failure of Cauchy and Lamé. It was one of the great calculations of the age and suitable reading for the final moments of a suicidal mathematician. Wolfskehl worked through the calculation line by line. Suddenly he was startled at what appeared to be a gap in the logic – Kummer had made an assumption and failed to justify a step in his argument. Wolfskehl wondered whether he had uncovered a serious flaw or whether Kummer's assumption was justified. If the former were true, then there was a chance that proving Fermat's Last Theorem might be a good deal easier than many had presumed.

He sat down, explored the inadequate segment of the proof, and became engrossed in developing a mini-proof which would either
consolidate Kummer's work or prove that his assumption was wrong, in which case all Kummer's work would be invalidated. By dawn his work was complete. The bad news, as far as mathematics was concerned, was that Kummer's proof had been remedied and the Last Theorem remained in the realm of the unattainable. The good news was that the appointed time of the suicide had passed, and Wolfskehl was so proud that he had discovered and corrected a gap in the work of the great Ernst Kummer that his despair and sorrow evaporated. Mathematics had renewed his desire for life.

Wolfskehl tore up his farewell letters and rewrote his will in the light of what had happened that night. Upon his death in 1908 the new will was read out, and the Wolfskehl family were shocked to discover that Paul had bequeathed a large proportion of his fortune as a prize to be awarded to whomsoever could prove Fermat's Last Theorem. The reward of 100,000 Marks, worth over £1,000,000 in today's money, was his way of repaying a debt to the conundrum that had saved his life.

The money was put into the charge of the
Königliche Gesellschaft der Wissenschaften
of Göttingen, which officially announced the competition for the Wolfskehl Prize that same year:

By the power conferred on us, by Dr. Paul Wolfskehl, deceased in Darmstadt, we hereby fund a prize of one hundred thousand Marks, to be given to the person who will be the first to prove the great theorem of Fermat.

The following rules will be followed:

(1) The
Königliche Gesellschaft der Wissenschaften
in Göttingen will have absolute freedom to decide upon whom the prize should be conferred. It will refuse to accept any manuscript written with the sole aim of entering the competition to obtain the Prize. It will only take into consideration those mathematical memoirs which have appeared in the form of a
monograph in the periodicals, or which are for sale in the bookshops. The Society asks the authors of such memoirs to send at least five printed exemplars.

(2) Works which are published in a language which is not understood by the scholarly specialists chosen for the jury will be excluded from the competition. The authors of such works will be allowed to replace them by translations, of guaranteed faithfulness.

(3) The Society declines responsibility for the examination of works not brought to its attention, as well as for the errors which might result from the fact that the author of a work, or part of a work, are unknown to the Society.

(4) The Society retains the right of decision in the case where various persons would have dealt with the solution of the problem, or for the case where the solution is the result of the combined efforts of several scholars, in particular concerning the partition of the Prize.

(5) The award of the Prize by the Society will take place not earlier than two years after the publication of the memoir to be crowned. The interval of time is intended to allow German and foreign mathematicians to voice their opinion about the validity of the solution published.

(6) As soon as the Prize is conferred by the Society, the laureate will be informed by the secretary, in the name of the Society; the result will be published wherever the Prize has been announced during the preceding year. The assignment of the Prize by the Society is not to be the subject of any further discussion.

(7) The payment of the Prize will be made to the laureate, in the next three months after the award, by the Royal Cashier of Göttingen University, or, at the receiver's own risk, at any other place he may have designated.

(8) The capital may be delivered against receipt, at the Society's will, either in cash, or by the transfer of financial values. The payment of the Prize will be considered as accomplished by the transmission of these financial values, even though their total value at the day's end may not attain 100,000 Marks.

(9) If the Prize is not awarded by 13 September 2007, no ulterior claim will be accepted.

The competition for the Wolfskehl Prize is open, as of today, under the above conditions.

Göttingen, 27 June 1908
Die Königliche Gesellschaft der Wissenschaften

It is worth noting that although the Committee would give 100,000 Marks to the first mathematician to prove that Fermat's Last Theorem is true, they would not award a single pfennig to anybody who might prove that it is false.

The Wolfskehl Prize was announced in all the mathematical journals and news of the competition rapidly spread across Europe. Despite the publicity campaign and the added incentive of an enormous prize the Wolfskehl Committee failed to arouse a great deal of interest among serious mathematicians. The majority of professional mathematicians viewed Fermat's Last Theorem as a lost cause and decided that they could not afford to waste their careers working on a fool's errand. However, the prize did succeed in introducing the problem to a whole new audience, a hoard of eager minds who were willing to apply themselves to the ultimate riddle and approach it from a path of complete innocence.

Ever since the Greeks, mathematicians have sought to spice up their textbooks by rephrasing proofs and theorems in the form of solutions to number puzzles. During the latter half of the nineteenth century this playful approach to the subject found its way into the popular press, and number puzzles were to be found
alongside crosswords and anagrams. In due course there was a growing audience for mathematical conundrums, as amateurs contemplated everything from the most trivial riddles to profound mathematical problems, including Fermat's Last Theorem.

Perhaps the most prolific creator of riddles was Henry Dudeney, who wrote for dozens of newspapers and magazines, including the
Strand, Cassell's
, the
Queen, Tit-Bits
, the
Weekly Dispatch
and
Blighty.
Another of the great puzzlers of the Victorian Age was the Reverend Charles Dodgson, lecturer in mathematics at Christ Church, Oxford, and better known as the author Lewis Carroll. Dodgson devoted several years to compiling a giant compendium of puzzles entitled
Curiosa Mathematica
, and although the series was not completed he did write several volumes, including
Pillow Problems.

The greatest riddler of them all was the American prodigy Sam Loyd (1841–1911), who as a teenager was making a healthy profit by creating new puzzles and reinventing old ones. He recalls in
Sam Loyd and his Puzzles: An Autobiographical Review
that some of his early puzzles were created for the circus owner and trickster P.T. Barnum:

Many years ago, when Barnum's Circus was of a truth ‘the greatest show on earth', the famous showman got me to prepare for him a series of prize puzzles for advertising purposes. They became widely known as the ‘Questions of the Sphinx', on account of the large prizes offered to anyone who could master them.

Strangely this autobiography was written in 1928, seventeen years after Loyd's death. Loyd passed his cunning on to his son, also called Sam, who was the real author of the book, knowing full well that anybody buying it would mistakenly assume that it had been written by the more famous Sam Loyd Senior.

Loyd's most famous creation was the Victorian equivalent of the Rubik's Cube, the ‘14–15' puzzle, which is still found in toyshops today. Fifteen tiles numbered 1 to 15 are arranged in a 4 × 4 grid, and the aim is to slide the tiles and rearrange them into the correct order. Loyd's offered a significant reward to whoever could complete the puzzle by swapping the ‘ 14' and ‘15' into their proper positions via any series of tile slides. Loyd's son wrote about the fuss generated by this tangible but essentially mathematical puzzle:

A prize of $1,000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of persons who say they performed the required feat. People became infatuated with the puzzle and ludicrous tales are told of shopkeepers who neglected to open their stores; of a distinguished clergyman who stood under a street lamp all through a wintry night trying to recall the way he had performed the feat. The mysterious feature of the puzzle is that none seem to be able to remember the sequence of moves whereby they feel sure they succeeded in solving the puzzle. Pilots are said to have wrecked their ships, and engineers rushed their trains past stations. A famous Baltimore editor tells how he went for his noon lunch and was discovered by his frantic staff long past midnight pushing little pieces of pie around on a plate!

Loyd was always confident that he would never have to pay out the $1,000 because he knew that it is impossible to swap just two pieces without destroying the order elsewhere in the puzzle. In the same way that a mathematician can prove that a particular equation has no solutions, Loyd could prove that his ‘14–15' puzzle is insoluble.