A picture of Pierre de Fermat inside a triangular object
Fermat's Enigma
Simon Singh

The fascinating story of how one of mathematics' hardest problems was solved. Takes a near-total survey of mathematical history, yet doesn't demand much knowledge of maths. I never thought a book about maths would be "hard to put down," but here we are.

Disclaimer: I am not a mathematician, so I've probably made a mistake in these notes. If you spot one, email me: hello (at) will patrick dot co dot uk.

'It is hard to conceive of any problem, in any discipline of science, so simply and clearly stated that could have withstood the test of advancing knowledge for so long.'

Fermat's last theorem (more accurately a conjecture because it's not a theorem until there's a proof) is that there are no integer (i.e. whole number) solutions to the following equation:

xn + yn = zn where n is an integer > 2.

Resisting the attempts of mathematicians to solve it for some 350 years, Fermat's conjecture was eventually solved by mathematics Professor Andrew Wiles in 1993-4. Unusually for mathematics, a field where collaboration is common and where patents are impossible, Wiles chose to develop his proof in secret.

Fermat's Enigma traces the story of mathematics from its earliest roots through the dark ages, the renaissance, the modern era, and Wiles' stunning breakthrough in the early nineties.

Pythagorean roots

xn + yn = zn appears, at first glance, to be remarkably similar to Pythagoras' theorem: x2 + y2 = z2.

Pythagoras is widely thought of as the first mathematician. Spending his early years gathering the known mathematical methods of the Babylonians and Egyptians, he settled in Croton where he founded the Pythagorean brotherhood.

The brotherhood was a secretive organisation sponsored by Milo of Croton, and thought of number as a religion. The brotherhood was, at first,  interested in excessive/abundant, defective and perfect numbers:

  • Excessive or abundant number: where the sum of the proper factors of n is greater than the number (e.g. the sum of the proper factors of 12 (1, 2, 3, 4, 6) is 16)
  • Defective or deficient number: where the sum of the proper factors of n is less than n (e.g. the sum of the proper factors of 25 (1, 5) is 6)
  • Perfect number: where the sum of the proper factors of n is equal to n (e.g. the sum of the proper factors of 6 (1, 2, 3) is 6)

Pythagoras discovered the connection between the physical world and the world of numbers, which he primarily did through the study of music. (We see this connection in many places, rivers being one extraordinary example. The actual length of a river is approximately πx, where x is the length of a straight line from the source of the river to the mouth.)

What was most interesting about Pythagoras was not that he discovered x2 + y2 = z2 but rather that he proved it.

A proof: a set of axioms arguing toward an irrefutable conclusion to create a theorem.

Scientific proof versus mathematical proof

Singh gives a powerful (if simplified) example of the difference between how scientists and mathematicians might approach a discrete problem:

'We have a chessboard with the two opposing corners removed, so that there are only 62 squares remaining. Now we take 31 dominoes shaped such that each domino covers exactly two squares. The question is: Is it possible to arrange the 31 dominoes so that they cover all the 62 squares on the chessboard?' (Page 23)

The scientific method might be to try and find an arrangement using the physical movement of the dominoes themselves. They may try repeatedly and not find anything, and have to settle for a reasonable level of probability.

A mathematician would instead concern themselves with finding the proof for whether or not this is possible at all. Anything less than this would be unacceptable.

A short note on computers: with the advent of computing after the Second World War, why not just use computers to solve n for as many numbers as you possibly can? Well, there are an infinite number of numbers. So even if you solve several million, or several billion, you've still got infinity to go. So while we did get up to solving n to four million by the 1980s, it's still not something that will make a difference.

Pythagoras killed, his secrets spread

When Croton was sacked, Pythagoras was murdered and his disciples fled. They took their secrets with them, and spread them around the known world. One of their primary interests at the time was the search for "Pythagorean triples."

A Pythagorean triple takes the form a2 + b2 = c2 where the squares of a and b equal the square of c. An example would be 3, 4, & 5: the squares of 3 and 4 are 9 and 16 which, when added together, are the square of 5: 25.

However, this is where the fun starts. When we change the power, e.g. to a cube, there don't appear to be any integer solutions to this problem.

Fermat went further, saying there were none at all. This was his conjecture, found among his notes after his death, to which no proof was provided.

The great library and the innocuous note in the margin

'In 1742, almost a century after Fermat's death, the Swiss mathematician Leonhard Euler asked his friend Clêrot to search Fermat's house in case some vital scrap of paper still remained. No clues were ever found as to what Fermat's proof might have been.' (Page 32)

Fermat scribbled his conjecture in the margin of a book, with no proof in sight. How that book came to be in his possession, and how it came to be published again with his notes, took more than a few twists of fate.

The Library of Alexandria, established by Ptolemy in roughly the third century BC, became what is widely regarded as the world's first university. Scouring the world for books, its librarians would confiscate the books of visitors, copy them, then give the copies back to the visitor.

As this great library grew, academics followed closed behind. The first head of the mathematics department was none other than Euclid. Euclid was interested in irrational numbers, those that are not expressable as an integer or a fraction. Common examples of these are π or √2. Writing in Elements he also laid the foundations of the study of geometry.

One of the books that survived the eventual destruction of the library at Alexandria was Arithmetica by Diophantus. A greek mathematician, not much is known about him. But he was chiefly concerned with number theory.

Diophantus discovered how, unusually, the number 26 was the only number between a square (25, so 52) and a cube (27, so 33). He also discovered "friendly numbers," where the divisors of a pair of numbers sum to the total of the other number. 220 and 284 do this, and Fermat also discovered 17,296 and 18,416.

It was in the margin of a copy of Arithmetica that Fermat made this observation:

"Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere."

Or, translated:

"It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers." (Page 61)

However, this wasn't all he wrote. He also noted the he had "a truly marvelous demonstration of this proposition which this margin is too narrow to contain.' (Page 62)

Fermat claimed he had a proof but left no clue to it. In addition, over the years, his other conjectures left in the margins of this book were proved one by one. This conjecture was the last one remaining and refused to be solved, hence the name "Fermat's Last Theorem."

'The fame of Fermat's Last Theorem comes solely from the sheer difficulty of proving it.' (Page 67)

The first breakthrough: Euler

'Euler had such an incredible intuition and vast memory that it was said he could map out the entire bulk of a calculation in his head without having to put pen to paper. Across Europe he was referred to as "analysis incarnate," and the French academician François Arago said, "Euler calculated without apparent effort as men breathe, or as eagles sustain themselves in the wind."' (Page 73)

Leonhard Euler was an 18th Century Swiss mathematician, famous for his creation of the algorithm. (The first of which was used to solve the "three body problem" to help ships navigate with accuracy, and was bought from Euler by the British Admiralty.)

He was the first to prove n=3 using Fermat's "method of infinite descent." 3, being a prime, suggested the possibility that, perhaps, you might be able to solve for all primes.

This was something that Sophie Germain, a late 18th and early 19th Century French mathematician, was able to do for primes p where 2p+1 also equal a prime. This is also called a "safe prime." (This allowed two other French mathematicians to go on to solve for n=5.)

The Wolfskehl Prize

After his affections were spurned by a woman he was deeply infatuated with, the 19th Century German physician and hobbyist mathemtician Paul Wolfskehl planned to commit suicide.

However, with time to spare before his scheduled moment of suicide at midnight, he spent time reading a proof in a paper by Ernst Kummer who had found a flaw in the conjecture of another mathematician, one who erroneously believed they had made progress in solving Fermat.

After noticing the flaw, he spent all night finishing the proof. When he was done, the sun had risen. Thankful that mathematics had saved him from taking his life, he decided to leave a prize of 100,000 Marks to the person who could finally solve Fermat. The rules of the prize dictated it would run from 1908 to 2007, and it prompted many amateurs to take up the challenge.

Axioms and strange loops

David Hilbert, a mathematician working around the turn of the century, began to establish a series of axioms upon which the rest of mathematics could be based. His argument was that all problems could be solved with "axiomatic set theory."

Several decades later, Kurt Gödel upset Hilbert's apple cart.

'By his early twenties Gödel had established himself in the mathematics department, but along with his colleagues he would occasionally wander down the corridor to attend meetings of the Wiener Kreis (Viennese Circle), a group of philosophers who would gather to discuss the day's great questions of logic. It was during this period that Gödel developed the ideas that would devastate the foundations of mathematics.' (Page 139)

The death of Hilbert's axiomatic set theory came in the shape of two of Gödel's theorems:

1st theorem of undecidability: "If axiomatic set theory is consistent, there exist theorems that can neither be proved or disproved." That is, there will always be problems that mathematics cannot answer.

2nd theorem of undecidability: "There is no constructive procedure that will prove axiomatic theory to be consistent." That is, you can never be sure that your choice of axioms won't lead to an inconsistency somewhere.

In the mid-60s, mathematician Paul Cohen's work on undecidable problems (including one of Hilbert's 23 axioms) led many to ask: what if Fermat was undecidable, and therefore could never be proved?

Elliptic curves and modular forms

Wiles' study as a graduate was on the subject of elliptic curves. It would be the knowledge he would later need to solve Fermat.

'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."' (Page 165)

In a completely different field of mathematics altogether, there is something called modular forms. I don't have the understanding of mathematics to explain them here, but they are essentially hyper symmetrical objects that exist on 4 axes in real and imaginary space. They are, in short, weird.

Two post-war Japanese mathematicians Yutaka Taniyaka and Goro Shimura made the conjecture that all elliptic curves were modular, not just some of them (as was already known).

'This was a doubly profound discovery. First, it suggested that deep down there was a fundamental  relationship between the modular form and the elliptic equation, objects that come from opposite ends of mathematics. Second, it meant that mathematicians, who already knew the M-series for the modular form, would not have to calculate the E-series for the corresponding elliptic equation because it would be the same as the M-series. Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth that enriches both subjects' (Page 183)

Joining the elliptic and modular worlds means problems in one can be solved in the other.

'If the Taniyama-Shimura conjecture was true it would enable mathematicians to tackle elliptic problems that had remained unsolved for centuries by approaching them through the modular world. The hope was that the fields of elliptic equations and modular forms could be unified. The conjecture also inspired the hope that links might exist between various other mathematical subjects' (Page 192)

Frey enters the fray

In 1985, German mathematician Gerhard Frey pointed out that it was possible to convert Fermat's Last Theorem into an elliptic equation. It appeared to be an elliptic equation so unusual that many thought it could not be modular.

This lead to the following incredible argument:

  1. 'If the Taniyama-Shimura conjecture can be proved to be true, then every elliptic equation must be modular.
  2. If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist.
  3. If the Frey elliptic equation does not exist, then there can be no solutions to Fermat's equation.
  4. Therefore Fermat's Last Theorem is true!' (Page 197)

The argument works both ways of course, meaning it could also mean that Fermat is false. But there was one remaining question before anyone could make any more progress: was it so strange an elliptic equation that it couldn't be modular? 18 months later, Ken Ribet at UC Berkeley had cracked it: Frey's equation was not modular.

'Fermat's Last Theorem was now inextricably linked to the Taniyama-Shimura conjecture. If somebody could prove that every elliptic equation is modular, then this would imply that Fermat's equation had no solutions, and immediately prove Fermat's Last Theorem.' (Page 202)

Wiles gets to work

"The romance of Fermat which had held me all my life was now combined with a problem that was professionally acceptable" (Page 206)

Wiles completely shut himself off, as much as was possible, from the regular mathematical world. He attended no conferences, no symposia, and removed himself from as much faculty work as possible. To avoid the appearance of not doing any work, he slowly published work he had already completed, piece by piece.

He reached a state of complete isolation and secrecy, giving him the focus and privacy he needed to do his work. This is an unusual mode for a mathematician, a group normally very collaborative. But Wiles feared that the attention that his work would attract would hold him back.

After years of work, and by using something called "Galois group theory" he had proven the first half of the problem. Struggling to find a way to complete the rest of it, he attended a single conference to catch up with his contemporaries to learn what the latest news was.

It was here that his old doctoral supervisor, John Coates, told him about the "Kolyvagin-Flach" method. This would prove to be the pivotal method that would allow him to continue on towards a proof.

In the very last year of his study, he divulged the secret to one colleague: Nick Katz. Katz was an expert in one particular method that Wiles needed help in understanding to solve the one remaining problem he needed to solve. Because it would take so long to talk through it all, they instead settled upon the plan of delivering a series of lectures.

Only available to graduate students, Nick Katz attended each lecture to check through all of the work that Wiles had done. As the subject matter was dense and had no particular point that it was driving towards, the graduate students quickly vanished leaving only Katz behind.

Katz, by the end, was convinced it worked. Wile's final problem fell. And nobody was any the wiser.

He had done it.

The proof

Under the inconspicuous title of "Modular Forms, Elliptic Curves and Galois Representations," Wiles delivered a series of three short lectures at his alma mater in Cambridge during a conference in 1993. After an intense rumour mill and incredible tension, Wiles began to deliver the lectures.

What was particularly incredible was that many of the people who had contributed toward the proof were there, in the room.

'"There was a typical dignified silence while I read out the proof and then I just wrote up the statement of Fermat's Last Theorem. I said, 'I think I'll stop here," and then there was sustained applause."' (Page 249)
"The proof of the Taniyama-Shimura conjecture was a far more important achievement than the solution of Fermat's Last Theorem, because it had immense consequences for many other mathematical theorems." (Page 251)

The final hurdle

Not content with the usual peer review of 2 to 3 of his peers, Wiles' proof had six reviewers assigned to it. It was a massive, 200 page manuscript that took an entire summer to review. The reviewers would email in their questions to the problems they would find, and Wiles would answer them one by one.

But during that summer, Nick Katz (who had been assigned as a reviewer) found a problem that Wiles could not easily answer. There was a fundamental flaw with his use of the Kolyvagin-Flach method.

He needed something to shore up this problem to make sure the proof was absolute, and it ended up being almost another year and a half for Wiles as he searched for the solution. Eventually, just as he was on the cusp of defeat, and as he was about to give up and publish his paper without a completed proof, he realised that he could return to his previously-abandoned Iwasawa Method, joining it with Kolyvagin-Flach.

'As he recounted these moments, the memory was so powerful that he was moved to tears: "It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes... The last fourteen months had been the most painful, humiliating, and depressing period of his mathematical career. Now one brilliant insight had brought an end to his suffering' (Page 275)
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