The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time here.
If the dream of every fledgling scientist is to make a discovery that will change the course of history, then the nightmare is that, having made the announcement and hit the headlines, someone discovers that you made a mistake and you will face ruin. This is what happened to Andrew Wiles in 1993.
For seven years he had worked secretly in his attic, attempting to solve a numerical puzzle that had defeated the greatest mathematicians for 350 years. He finally cracked it, or so he thought, and described the proof in 3 hours of lectures, during which no one noticed a crucial flaw. The world’s leading media headlined his achievement; he was invited to endorse a brand of jeans. Overnight, mathematics and the mystical power of numbers had suddenly become fashionable. But then the problems began.
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Wiles’s proof involved hundreds of individual logical steps, carefully joined one to the next. Only later, as experts examined the structure, did they realise that the link connecting two of those steps was flawed. Only one weak link in a huge chain, admittedly, but enough to break the entire structure.
Thankfully the story had a happy ending, as Wiles’s endurance and ingenuity perfected that link and finally proved “Fermat’s last theorem”. The beauty of this theorem is that it is supremely easy to understand. Every school child learns Pythagoras’s rule for right-angled triangles, “The square of the hypotenuse is equal to the sum of the squares of the other two sides”, and can demonstrate that simple numbers satisfy this statement, such as 32 + 42 = 52. This provides the inspiration for Fermat’s problem that thwarted three centuries of mathematicians: namely, that there are no whole-number integers that satisfy the equation an + bn = cn for n greater than 2. For example, try finding three integers a, b and c that are solutions to a3 + b3 = c3. But don’t spend too long: there aren’t any.
The tantalising feature is that Fermat, who first stated this theorem, wrote that he had “discovered a truly marvellous proof of this, which, however the margin is not large enough to contain”. What this proof was has never been found, and there is doubt that Fermat had actually achieved what he thought.
Wiles first encountered Fermat’s last theorem in a library book at the age of 10 and decided to devote his life to it. It was at about that same age that I first read about the theorem in a newspaper article. However, unknown to me and the editor, the typesetter had made a misprint – the real theorem an + bn = cn had been mis-transcribed as an × bn = cn. I recall the sense of wonder as I realised that, if a × b = c, the equation would have a solution; at 10 years old one has the confidence to believe that one can see in an instant what has evaded the greatest mathematicians for 300 years. I wrote to the editor, who kindly informed me of the typing error. I was amazed that the simple replacement of a multiplication sign by addition could turn a trivial question into one so profound. I quit then; Wiles kept the dream alive.
It was 1637 when Pierre de Fermat, a lawyer and mathematician, wrote his renowned marginal comment. It is hard to think of any problem so simply and clearly stated that it could withstand the test of advancing knowledge for so long. In the 360 years from the time the theorem was formulated to when it was proven, we progressed from an Earth-centred universe to a big bang cosmology, from ignorance to the threshold of a grand unified theory of physics. Yet Fermat’s last theorem, long regarded as the Himalayan peak of number theory, remained unconquered. Academies and wealthy patrons began to offer financial prizes for a complete proof.
Meanwhile, new areas of mathematics were being developed, among them group theory by mathematician Évariste Galois. He wrote down his discoveries of two types of equations, called elliptic curves and modular functions, in 1832, on the night before he was killed in a duel.
An important piece of the story occurred in 1955, when mathematicians Yutaka Taniyama and Goro Shimura conjectured that there is a bridge between Galois’s two apparently unrelated mathematical fields of modular functions and elliptic curves. They were unable to complete the proof, however. Taniyama died soon thereafter and did not see the triumphant climax to the conjecture that bears his name.
The critical link that led to Wiles’s assault on the summit happened in 1984, when Ken Ribet showed that if the Shimura-Taniyama conjecture were true, then Fermat’s last theorem would be also. Ribet completed his proof following an after-dinner remark by his colleague, Barry Mazur. Famously, Ribet looked at Mazur, then back at his half-drunk cappuccino and then returned his gaze to Mazur with disbelief.
The way was now set for Andrew Wiles. If he could prove the Shimura-Taniyama conjecture then he would, thanks to Ribet’s work, have also proved Fermat. This led to seven years’ work in secret isolation, his moment of triumph when he announced his success, followed by months of agony while he tried to put his broken theory right. Finally, in 1995, his triumph was complete.
During Wiles’s eight-year ordeal he had incorporated nearly all the advances in 20th-century number theory into his final proof. He had combined traditional mathematical methods with ones created by his contemporaries, and with his own newly minted techniques. Previously disparate areas of mathematics had been brought together to complete the proof of Fermat’s last theorem. Wiles’s proof revolutionised several areas of mathematics and runs to over 100 pages. It is hard to believe that in 1637 Fermat could really have proved it and relegated it to a marginal comment.
While Wiles was toiling with his broken proof in 1994, and all the world knew, a mathematical graffito appeared on the 8th Street subway station in New York City parodying Fermat’s original. The artwork announced, “an + bn = cn ; no solutions. I have found a marvellous proof, but I cannot write it here as my train is coming.”
The elegant simplicity of Fermat’s theorem, when married with the almost perfect impenetrability of its proof, is testament to our mystical fascination with integers. Thanks to Pythagoras, the simple addition of integers, a + b = c, was elevated to addition of their squares: a2 + b2 = c2. There is an infinity of integers that satisfy the first of these, and likewise there is an infinity that satisfy the sums of the squares. Mathematicians love the symmetries of numbers, and long ago looked for examples where the cubes of non-zero integers add together: a3 + b3 = c3. But as Fermat suspected and eventually Wiles proved, there are no integers that satisfy that or any higher power. Addition of integers ends with squares.
Or, at least, it does until and unless someone finds a counter-example – and in an episode of the cartoon television show The Simpsons, it seemed that Fermat had been upended. Homer Simpson, in a dream, wrote that 1782 to the 12th power plus 1841 to the 12th power equals 1922 to the 12th power: 178212Ěý + 184112 = 192212.
An article in the San Francisco Chronicle in November 2005 featured this remarkable equation and that the numbers do indeed match. With a pocket calculator, you can verify the equality for yourself. Have we found the long-sought refutation of Fermat?
According to the calculator, the answer would appear to be yes. But what we have really demonstrated is the limitation of a calculator, not of Fermat. What’s more, it’s possible to see the equation isn’t true without computing the individual terms. The product of evens (the first term) is even; the product of odds (the second term) is odd. The sum of an even and odd is odd (the left-hand side of the equation). As for the right-hand side, a product of even powers, it must be even. Something is obviously awry here.
The explanation is that the 12th power leads to numbers with forty integers, whereas after about ten digits, most hand calculators round the final digit up or down to keep an approximation to the answer. In reality, the sum on the left of the equation equals the 12th power not of 1922 but of 1921.99999996, which on a hand calculator rounds up to 1922. That may be fine for practical computation – but not for the perfect precision required of a mathematical theorem.
Wiles’s proof of Fermat’s theorem survives! He completed his proof shortly after Tom Stoppard’s play Arcadia opened in London’s West End. A character in the play says that Fermat had no proof but claimed otherwise to “torment later generations”. It’s possible that Stoppard was right.
The editors at the San Francisco Chronicle were among those destined to be tormented. After their story about Homer Simpson’s dream, all would have been well had they not added the assurance that the sums truly matched. They were inundated with letters from mathematical savants and number nerds explaining why Homer Simpson’s dream had become the Chronicle’s nightmare. On 16 December 2005, reporter Dick Rogers wrote a , wittily echoing Fermat’s original note: “There’s not enough room in this column to credit readers with all the course-correcting feedback they give.”
