Fermat's Last Theorem

by Simon Singh

  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.

  Figure 12. By sliding the tiles it is possible to create various disordered arrangements. For each arrangement it is possible to measure the amount of disorder, via the disorder parameter Dp.

  Loyd’s proof began by defining a quantity which measured how disordered a puzzle is, the disorder parameter Dp. The disorder parameter for any given arrangement is the number of tile pairs which arc in the wrong order, so for the correct puzzle, as shown in Figure 12(a), Dp = 0, because no tiles are in the wrong order.

  By starting with the ordered puzzle and then sliding the tiles around, it is relatively easy to get to the arrangement shown in Figure 12(b). The tiles are in the correct order until we reach tiles 12 and 11. Obviously the 11 tile should come before the 12 tile and so this pair of tiles is in the wrong order. The complete list of tile pairs which are in the wrong order is as follows: (12,11), (15,13), (15,14), (15,11), (13,11) and (14,11). With six tile pairs in the wrong order in this arrangment, Dp = 6. (Note that tile 10 and tile 12 are next to each other, which is clearly incorrect, but they are not in the wrong order. Therefore this tile pair does not contribute to the disorder parameter.)

  After a bit more sliding we get to the arrangement in Figure 12(c). If you compile a list of tile pairs in the wrong order then you will discover that Dp = 12. The important point to notice is that in all these cases, (a), (b) and (c), the value of the disorder parameter is an even number (0, 6 and 12). In fact, if you begin with the correct arrangement and proceed to rearrange it, then this statement is always true. As long as the empty square ends up in the bottom right-hand corner, any amount of tile sliding will always result in an even value for Dp. The even value for the disorder parameter is an integral property of any arrangement derived from the original correct arrangement. In mathematics a property which always holds true no matter what is done to the object is called an invariant.

  However, if you examine the arrangement which was being sold by Loyd, in which the 14 and 15 were swapped, then the value of the disorder parameter is one, Dp = 1, i.e. the only pair of tiles out of order are the 14 and 15. For Loyd’s arrangement the disorder parameter has an odd value! Yet we know that any arrangement derived from the correct arrangement has an even value for the disorder parameter. The conclusion is that Loyd’s arrangement cannot be derived from the correct arrangement, and conversely it is impossible to get from Loyd’s arrangement back to the correct one – Loyd’s $1,000 was safe.

  Loyd’s puzzle and the disorder parameter demonstrate the power of an invariant. Invariants provide mathematicians with an important strategy to prove that it is impossible to transform one object into another. For example, an area of current excitement concerns the study of knots, and naturally knot theorists are interested in trying to prove whether or not one knot can be transformed into another by twisting and looping but without cutting. In order to answer this question they attempt to find a property of the first knot which cannot be destroyed no matter how much twisting and looping occurs – a knot invariant. They then calculate the same property for the second knot. If the values are different then the conclusion is that it must be impossible to get from the first knot to the second.

  Until this technique was invented in the 1920s by Kurt Reidemeister it was impossible to prove that one knot could not be transformed into any other knot. In other words before knot invariants were discovered it was impossible to prove that a granny knot is fundamentally different from a reef knot, an overhand knot or even a simple loop with no knot at all. The concept of an invariant property is central to many other mathematical proofs and, as we shall see in Chapter 5, it would be crucial in bringing Fermat’s Last Theorem back into the mainstream of mathematics.

  By the turn of the century, thanks to the likes of Sam Loyd and his ‘14–15’ puzzle, there were millions of amateur problem-solvers throughout Europe and America eagerly looking for new challenges. Once news of Wolfskehl’s legacy filtered down to these budding mathematicians, Fermat’s Last Theorem was once again the world’s most famous problem. The Last Theorem was infinitely more complex than even the hardest of Loyd’s puzzles, but the prize was vastly greater. Amateurs dreamed that they might be able to find a relatively simple trick which had eluded the great professors of the past. The keen twentieth-century amateur was to a large extent on a par with Pierre de Fermat when it came to knowledge of mathematical techniques. The challenge was to match the creativity with which Fermat used his techniques.

  Within a few weeks of announcing the Wolfskehl Prize an avalanche of entries poured into the University of Göttingen. Not surprisingly all the proofs were fallacious. Although each entrant was convinced that they had solved this centuries-old problem they had all made subtle, and sometimes not so subtle, errors in their logic. The art of number theory is so abstract that it is frighteningly easy to wander off the path of logic and be completely unaware that one has strayed into absurdity. Appendix 7 shows the sort of classic e
rror which can easily be overlooked by an enthusiastic amateur.

  Regardless of who had sent in a particular proof, every single one of them had to be scrupulously checked just in case an unknown amateur had stumbled upon the most sought after proof in mathematics. The head of the mathematics department at Göttingen between 1909 and 1934 was Professor Edmund Landau and it was his responsibility to examine the entries for the Wolfskehl Prize. Landau found that his research was being continually interrupted by having to deal with the dozens of confused proofs which arrived on his desk each month. To cope with the situation he invented a neat method of off-loading the work. The professor printed hundreds of cards which read:

  Landau would then hand each new entry, along with a printed card, to one of his students and ask them to fill in the blanks.

  The entries continued unabated for years, even following the dramatic devaluation of the Wolfskehl Prize – the result of the hyperinflation which followed the First World War. There are rumours which say that anyone winning the competition today would hardly be able to purchase a cup of coffee with the prize money, but these claims are somewhat exaggerated. A letter written by Dr F. Schlichting, who was responsible for dealing with entries during the 1970s, explains that the prize was then still worth over 10,000 Marks. The letter, written to Paulo Ribenboim and published in his book 13 Lectures on Fermat’s Last Theorem, gives a unique insight into the work of the Wolfskehl committee:

  Dear Sir,

  There is no count of the total number of ‘solutions’ submitted so far. In the first year (1907–1908) 621 solutions were registered in the files of the Akademie, and today they have stored about 3 metres of correspondence concerning the Fermat problem. In recent decades it was handled in the following way: the secretary of the Akademie divides the arriving manuscripts into:

  (1) complete nonsense, which is sent back immediately,

  (2) material which looks like mathematics.

  The second part is given to the mathematical department, and there the work of reading, finding mistakes and answering is delegated to one of the scientific assistants (at German universities these are graduated individuals working for their Ph.D.) – at the moment I am the victim. There are about 3 or 4 letters to answer each month, and this includes a lot of funny and curious material, e.g. like the one sending the first half of his solution and promising the second if we would pay 1,000 DM in advance; or another one, who promised me 1% of his profits from publications, radio and TV interviews after he got famous, if only I would support him now; if not, he threatened to send it to a Russian mathematics department to deprive us of the glory of discovering him. From time to time someone appears in Göttingen and insists on personal discussion.

  Nearly all ‘solutions’ are written on a very elementary level (using the notions of high school mathematics and perhaps some undigested papers in number theory), but can nevertheless be very complicated to understand. Socially, the senders are often persons with a technical education but a failed career who try to find success with a proof of the Fermat problem. I gave some of the manuscripts to physicians who diagnosed heavy schizophrenia.

  One condition of Wolfskehl’s last will was that the Akademie had to publish the announcement of the prize yearly in the main mathematical periodicals. But already after the first years the periodicals refused to print the announcement, because they were overflowed by letters and crazy manuscripts.

  I hope that this information is of interest to you.

  Yours sincerely,

  F. Schlichting

  As Dr Schlichting mentions, competitors did not restrict themselves to sending their ‘solutions’ to the Akademie. Every mathematics department in the world probably has its cupboard of purported proofs from amateurs. While most institutions ignore these amateur proofs, other recipients have dealt with them in more imaginative ways. The mathematical writer Martin Gardner recalls a friend who would send back a note explaining that he was not competent to examine the proof. Instead he would provide them with the name and address of an expert in the field who could help – that is to say, the details of the last amateur to send him a proof. Another of his friends would write: ‘I have a remarkable refutation of your attempted proof, but unfortunately this page is not large enough to contain it.’

  Although amateur mathematicians around the world have spent this century trying and failing to prove Fermat’s Last Theorem and win the Wolfskehl Prize, the professionals have continued largely to ignore the problem. Instead of building on the work of Kummer and the other nineteenth-century number theorists, mathematicians began to examine the foundations of their subject in order to address some of the most fundamental questions about numbers. Some of the greatest figures of the twentieth century, including Bertrand Russell, David Hilbert and Kurt Gödel, tried to understand the most profound properties of numbers in order to grasp their true meaning and to discover what questions number theory can and, more importantly, cannot answer. Their work would shake the foundations of mathematics and ultimately have repercussions for Fermat’s Last Theorem.

  The Foundations of Knowledge

  For hundreds of years mathematicians had been busy using logical proof to build from the known into the unknown. Progress had been phenomenal, with each new generation of mathematicians expanding on their grand structure and creating new concepts of number and geometry. However, towards the end of the nineteenth century, instead of looking forward, mathematical logicians began to look back to the foundations of mathematics upon which everything else was built. They wanted to verify the fundamentals of mathematics and rigorously rebuild everything from first principles, in order to reassure themselves that those first principles were reliable.

  Mathematicians are notorious for being sticklers when it comes to requiring absolute proof before accepting any statement. Their reputation is clearly expressed in a story told by Ian Stewart in Concepts of Modern Mathematics:

  An astronomer, a physicist, and a mathematician (it is said) were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field. ‘How interesting,’ observed the astronomer, ‘all Scottish sheep are black!’ To which the physicist responded, ‘No, no! Some Scottish sheep are black!’ The mathematician gazed heavenward in supplication, and then intoned, ‘In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.’

  Even more rigorous than the ordinary mathematician is the mathematician who specialises in the study of mathematical logic. Mathematical logicians began to question ideas which other mathematicians had taken for granted for centuries. For example, the law of trichotomy states that every number is either negative, positive or zero. This seems to be obvious and mathematicians had tacitly assumed it to be true, but nobody had ever bothered to prove that this really was the case. Logicians realised that, until the law of trichotomy had been proved true, then it might be false, and if that turned out to be the case then an entire edifice of knowledge, everything that relied on the law, would collapse. Fortunately for mathematics, at the end of the last century the law of trichotomy was proved to be true.

  Ever since the ancient Greeks, mathematics had been accumulating more and more theorems and truths, and although most of them had been rigorously proved mathematicians were concerned that some of them, such as the law of trichotomy, had crept in without being properly examined. Some ideas had become part of the folklore and yet nobody was quite sure how they had been originally proved, if indeed they ever had been, so logicians decided to prove every theorem from first principles. However, every truth had to be deduced from other truths. Those truths, in turn, first had to be proved from even more fundamental truths, and so on. Eventually the logicians found themselves dealing with a few essential statements which were so fundamental that they themselves could not be proved. These fundamental assumptions are the axioms of mathematics.

  One example of the axioms is the commutative law of addition, which simply stat
es that, for any numbers m and n,

  m + n = n + m

  This and the handful of other axioms are taken to be self-evident, and can easily be tested by applying them to particular numbers. So far the axioms have passed every test and have been accepted as being the bedrock of mathematics. The challenge for the logicians was to rebuild all of mathematics from these axioms. Appendix 8 defines the set of arithmetic axioms and gives an idea of how logicians set about building the rest of mathematics.

  A legion of logicians participated in the slow and painful process of rebuilding the immensely complex body of mathematical knowledge using only a minimal number of axioms. The idea was to consolidate what mathematicians thought they already knew by employing only the most rigorous standards of logic. The German mathematician Hermann Weyl summarised the mood of the time: ‘Logic is the hygiene the mathematician practises to keep his ideas healthy and strong.’ In addition to cleansing what was known, the hope was that this fundamentalist approach would also throw light on as yet unsolved problems, including Fermat’s Last Theorem.

  The programme was headed by the most eminent figure of the age, David Hilbert. Hilbert believed that everything in mathematics could and should be proved from the basic axioms. The result of this would be to demonstrate conclusively the two most important elements of the mathematical system. First, mathematics should, at least in theory, be able to answer every single question – this is the same ethos of completeness which had in the past demanded the invention of new numbers like the negatives and the imaginaries. Second, mathematics should be free of inconsistencies – that is to say, having shown that a statement is true by one method, it should not be possible to show that the same statement is false via another method. Hilbert was convinced that, by assuming just a few axioms, it would be possible to answer any imaginable mathematical question without fear of contradiction.