Fermat's Last Theorem

• 1. Introduction

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  Fermat's Last Theorem began around 1637 as a note in the margin of a book. Pierre de Fermat was reading his copy of Arithmetica when he wrote down a claim about whole numbers and their powers. He then added that he had found a proof, but that it was too large to fit in the margin. Nobody has ever found that proof. The claim stayed open for centuries. It drew in prize committees, generations of professional mathematicians, and a long line of amateurs with confident solutions. This documentary asks whether Fermat could really have been right, what the long effort to settle his note built along the way, and how the answer finally arrived from a direction almost no one expected.

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• 2. Triples and Powers

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  The numbers 3, 4 and 5 describe a triangle with a right angle. Builders and early geometers knew that any triangle with sides in the ratio 3 to 4 to 5 contains one. The numbers 5, 12 and 13 do the same job. Whole numbers where two squares add up to a third square are called Pythagorean triples, a name that honours the ancient Greek Pythagoras. There are infinitely many of them. Babylonian, Greek, Chinese and Indian mathematicians all studied ways of generating them.

  With an exponent of 2, positive whole numbers fit Fermat's pattern in endless variety, and so they do with an exponent of 1. That had been known since antiquity. Fermat's note said something starker. Once the exponent is any whole number above 2, no positive whole numbers work at all.

  Diophantus, a mathematician from 3rd-century Alexandria, wrote the book Fermat was reading. He studied equations that call for whole-number answers, and they are now called Diophantine equations in his honour. A typical problem asks for two integers when only their sum and the sum of their squares are known. Such puzzles are ancient. The Babylonians solved the quadratic kind around 1800 BC, and the Euclidean algorithm, from about the 5th century BC, handles the linear kind. Only part of Arithmetica survived, and Fermat knew it through a new Latin translation published in 1621 by Claude Bachet.

  Problem II.8 of Arithmetica asks how to split a given square number into two other squares. Diophantus showed how to do it. Fermat wrote his note right beside that problem, and it came to light only some thirty years later, after his death.

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• 3. A Doubtful Proof

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  Fermat died in 1665, and in 1670 his son Clément-Samuel Fermat published a new edition of Arithmetica with his father's comments added. That is how the marginal note reached the public. Strictly speaking it was no theorem, because a theorem needs a proof. It earned the name Last Theorem because it was the last of Fermat's asserted theorems still unproved.

  His theorem on sums of two squares was one of several claims that Fermat made without proof and that others later established. The marginal note refused to follow them. As the decades passed, doubt grew that Fermat had ever held a correct proof, and the statement came to be called a conjecture instead.

  Only one related proof by Fermat has survived, and it concerns the exponent 4. He used a technique called infinite descent to show that the area of a right triangle with whole-number sides can never equal the square of a whole number.

  Fermat challenged his correspondents Marin Mersenne, Blaise Pascal and John Wallis with particular cases. He never posed the general case to them. In the last thirty years of his life he never again wrote of his "truly marvelous proof", and he never published it.

  Van der Poorten reads that silence as a sign that Fermat realised he had no proof. He quotes the mathematician Weil as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used are unknown.

  Harvey Friedman's grand conjecture implies that any provable theorem can be proved using only elementary function arithmetic. Yet such a proof could involve millions of steps, far too many to have been Fermat's. The proof that finally arrived relied on techniques from the 20th century, and a proof from Fermat's day would have had to look elementary beside it.

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• 4. Exponent by Exponent

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  A single proof, for the exponent 4, handed mathematicians a shortcut. It shows that if the theorem fails for an exponent that is not prime, it must also fail for a smaller exponent. So only the odd prime exponents remained to be investigated.

  Between 1637 and 1839, only three odd primes were conquered: 3, 5 and 7. In the 10th century, Abu-Mahmud Khojandi stated an early case, but his attempted proof was incorrect.

  In 1770, Leonhard Euler offered a proof for the exponent 3 using infinite descent. It contained a major gap. Euler had already proved the missing lemma in other work, so he is generally credited with the first proof.

  Around 1825, Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet independently proved the exponent 5. Gabriel Lamé then proved the exponent 7 in 1839. His argument was complicated, and Lebesgue simplified it in 1840.

  Mathematicians also proved the exponents 6, 10 and 14. Dirichlet published his proof for 14 in 1832, seven years before Lamé's proof for 7. Strictly these were unnecessary, because they follow from the cases of 3, 5 and 7. Their reasoning nevertheless differed from the odd-exponent proofs.

  All of the proofs for the exponents 3, 5, 7, 6, 10 and 14 leaned on Fermat's descent, in its original form or in disguised forms such as descent on elliptic curves. The details were tied to each exponent and grew more complicated as the prime grew. That made it unlikely that the general theorem could ever be built from individual cases. Niels Henrik Abel and Peter Barlow published some general results in the early 19th century, but the first significant work on the general theorem came from Sophie Germain.

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• 5. Germain and Kummer

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  Sophie Germain's plan, set out in the early 19th century, used auxiliary primes built from each prime exponent. She hoped to prove by induction that infinitely many such primes must divide the product of the three numbers in any solution. That product has only finitely many prime factors, so success would have proved the theorem. She developed many techniques but never reached that goal.

  In 1847, Gabriel Lamé outlined a proof that factored the equation using complex numbers built from the roots of the number 1. It failed because he assumed these numbers factor uniquely into primes, as ordinary integers do. Joseph Liouville spotted the gap immediately. He later read a paper by Ernst Kummer showing that unique factorisation does fail.

  Kummer then asked whether new primes could be added so that unique factorisation was restored. He succeeded by developing what he called ideal numbers. Following Lamé's general approach, he proved the theorem for all regular primes. The irregular primes defeated him, and they are conjectured to make up about 39 percent of all primes. Eleven of them lie below 270, beginning with 37, 59 and 67 and ending with 257 and 263.

  A popular story says Kummer was led to ideal numbers by Fermat's theorem, and that he believed he had proved it until Dirichlet pointed out the flaw. Kurt Hensel first told that story in 1910, and the evidence suggests it comes from a confusion by one of his sources. Harold Edwards called the belief that Kummer was mainly interested in the theorem "surely mistaken".

  In the 1920s, Louis Mordell posed a conjecture implying that for any exponent above 2, the equation has at most finitely many nontrivial primitive whole-number solutions. Gerd Faltings proved it in 1983, and it is now called Faltings' theorem.

  In 1954, Harry Vandiver used a SWAC computer to prove the theorem for all primes up to 2521. By 1978, Samuel Wagstaff had reached every prime below 125,000. By 1993, the theorem had been proved for all primes below four million. Mathematicians knew from other conjectures, such as the one tied to Skewes' number, that a pattern can hold across an enormous range and then fail.

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• 6. An Unlikely Bridge

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  Around 1955, two Japanese mathematicians, Goro Shimura and Yutaka Taniyama, suspected a link between elliptic curves and modular forms. These are two apparently unrelated branches of mathematics. Their idea, now called the modularity theorem, says that every elliptic curve is modular, meaning it can be matched with a unique modular form.

  At first the link was dismissed as unlikely or highly speculative. André Weil then found evidence that supported it without proving it, and people began to take it seriously. The idea became known as the Taniyama-Shimura-Weil conjecture. Even so, it was widely seen as beyond reach.

  John Coates, who later supervised Andrew Wiles's doctorate, said the problem seemed "impossible to actually prove". He added, "I thought I probably wouldn't see it proved in my lifetime." Ken Ribet counted himself among "the vast majority of people who believed" it was completely inaccessible. He said that Andrew Wiles was probably one of the few people on earth with "the audacity to dream that you can actually go and prove" it.

  In 1984, Gerhard Frey noticed that a hypothetical solution to Fermat's equation would produce an elliptic curve with such unusual properties that it seemed unlikely to be modular. If that held, a disproof of Fermat's Last Theorem would disprove the modularity conjecture. A proof of modularity would then prove Fermat. Frey outlined the idea but did not prove it.

  Jean-Pierre Serre identified the missing piece, called the epsilon conjecture, and gave an almost complete proof of it. Ken Ribet finished the job in 1986. After that, proving the modularity conjecture for the class of curves called semistable would automatically prove the theorem for every exponent at once. It was the first route that promised to cover all numbers, not just some.

  When Ribet's result came out in 1986, it reached an English mathematician who had worked with elliptic curves and had loved Fermat's problem since childhood. His name was Andrew Wiles.

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• 7. The Secret Proof

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  For six years, Andrew Wiles worked on the modularity conjecture in near-total secrecy. He hid the effort by releasing earlier results in small, separate papers, and he confided only in his wife. His first significant breakthrough used Galois theory. Around 1990-91 he switched to extending Iwasawa theory, but by mid-1991 that approach also seemed to miss the central issues.

  Then he began asking colleagues for any hint of cutting-edge research. He found an Euler system, recently developed by Victor Kolyvagin and Matthias Flach, that seemed "tailor made" for the inductive part of his argument. He extended it, and it worked. The approach was new to mathematics and to him, so in January 1993 he asked his Princeton colleague Nick Katz to check his reasoning for subtle errors. They concluded that the techniques seemed to hold.

  By mid-May 1993, Wiles was ready to tell his wife he thought he had solved it. From the 21st to the 23rd of June 1993, he gave three lectures at the Isaac Newton Institute for Mathematical Sciences. He presented a proof of the conjecture for semistable elliptic curves, which together with Ribet's result implied Fermat's Last Theorem.

  On the 23rd of August 1993, Nick Katz, acting as a reviewer, alerted Wiles to an error. A critical step contained a mistake in a bound on the order of a particular group. The rest of Wiles's work remained significant and innovative. Without that step, though, there was no proof of Fermat's Last Theorem.

  Wiles spent almost a year trying to repair it, first alone and then with his former student Richard Taylor. By the end of 1993, rumours said the proof had failed. Colleagues pressed him to release his work, complete or not, so others could use it. The flaw had looked minor at first. It now seemed serious and hard to fix.

  On the morning of the 19th of September 1994, Wiles says, he was close to giving up and publishing so that others could try to fix the error. He was taking a last look at why the Kolyvagin-Flach approach failed. Then he saw that the very reason it failed meant his original Iwasawa attempt could be made to work. Neither approach was enough alone, but together they were powerful enough. He later said, "It was so indescribably beautiful; it was so simple and so elegant." He recalled, "I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes." Of the day that followed, he said, "Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there." He called it "the most important moment of my working life" and added, "Nothing I ever do again will mean as much."

  On the 24th of October 1994, Wiles submitted two manuscripts. One was titled "Modular elliptic curves and Fermat's Last Theorem". The other, "Ring theoretic properties of certain Hecke algebras", was co-written with Taylor and showed that the conditions needed for the corrected step were met. Both papers filled the entire May 1995 issue of the Annals of Mathematics. They established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.

  Diamond finished more of the conjecture in 1996, Conrad and colleagues in 1999, and Breuil and colleagues in 2001. Each built on Wiles's work to remove the remaining cases. The fully proved conjecture is now the modularity theorem. Wiles's method of identifying a deformation ring with a Hecke algebra, called an R=T theorem, has itself become an influential tool in algebraic number theory.

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• 8. Prizes and Pretenders

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  In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of the theorem. In 1857 it awarded Kummer 3,000 francs and a gold medal for his research on ideal numbers, although he had not entered the contest. The Academy of Brussels offered another prize in 1883.

  In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks to the Göttingen Academy of Sciences as a prize for a complete proof. The sum was large for its time. On the 27th of June 1908, the academy published nine rules. The proof had to appear in a peer-reviewed journal, the prize could not be paid until two years after publication, and no prize would be given after the 13th of September 2007. Wiles collected the money, then worth $50,000, on the 27th of June 1997.

  In March 2016, the Norwegian government awarded Wiles the Abel Prize, worth 600,000 euros. The award cited "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".

  Before Wiles, the Wolfskehl committee received thousands of incorrect proofs, amounting to roughly ten feet of correspondence. In the first year alone, 1907-1908, 621 attempts arrived. By the 1970s the rate had fallen to roughly 3-4 attempts per month. According to some claims, Edmund Landau used a preprinted form for these submissions, with the spot for the first mistake left blank for a graduate student to fill in.

  F. Schlichting, a Wolfskehl reviewer, said most of the proofs used elementary methods taught in schools. He said they often came from "people with a technical education but a failed career". The historian Howard Eves put it this way: "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."

  The theorem also found a niche in popular culture. In Arthur Porges' 1954 short story "The Devil and Simon Flagg", a mathematician bargains with the Devil that he cannot produce a proof within twenty-four hours. In a 1989 episode of Star Trek: The Next Generation, "The Royale", Captain Picard says the theorem is still unproven in the 24th century. The real proof arrived five years after the episode first aired.

  Simon Singh's 1997 book Fermat's Last Theorem became the first mathematics book to reach number one in the United Kingdom. The documentary The Proof, on which it was based, won a BAFTA award in 1997. In a 1998 episode of The Simpsons, "The Wizard of Evergreen Terrace", Homer Simpson writes an equation on a blackboard that seems to be a counterexample. It is wrong, yet it looks correct on a calculator that shows 10 significant figures.

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