Christopher: Most people think the greatest intellectual quests are about finding something useful, like a cure for a disease. But the world's most difficult puzzle, which took 358 years to solve, was famously, gloriously, and completely useless. And that’s exactly what made it so powerful. Lucas: That is such a wild thought. That the most profound challenge for the greatest minds on the planet for over three centuries had zero practical application. It’s like climbing Everest just because it’s there, but the mountain is made of pure logic and it takes generations to even find the base camp. Christopher: That's the paradox at the heart of the book we're diving into today: Fermat's Last Theorem by Simon Singh. Lucas: Right, and Singh is the perfect person to tell this story. He's not just a writer; he produced an award-winning documentary on this very topic. You can feel that cinematic, storytelling instinct on every page. He turns a math problem into a thriller. Christopher: He absolutely does. He understands that this isn't a story about equations; it's a story about obsession, genius, and heartbreak. And it all begins with a ten-year-old boy in a public library.

Christopher: The year is 1963. A ten-year-old boy named Andrew Wiles is browsing the shelves of his local library in Cambridge, England. He’s not looking for adventure stories; he’s looking for puzzle books. And he finds one with a chapter on a single problem, a problem that looked so deceptively simple. Lucas: What was the problem? Lay it on us. Christopher: It was an equation that almost everyone learns in school: Pythagoras's theorem, x² + y² = z². There are tons of whole number solutions for that, like 3, 4, and 5. But a 17th-century French mathematician, Pierre de Fermat, had scribbled a note in the margin of a book. He claimed that if you change the 'squared' to anything higher—cubed, or to the power of four, five, and so on—the equation xⁿ + yⁿ = zⁿ has zero whole number solutions. Lucas: Okay, so no solutions exist for any power greater than two. That’s the whole thing? Christopher: That's the whole thing. And then Fermat added the most infuriating, tantalizing line in the history of science. He wrote, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Lucas: Oh, come on! That’s the ultimate intellectual troll. That’s like tweeting "Just solved the meaning of life, but my phone's about to die." Christopher: Exactly. And for 358 years, nobody could find his "marvelous proof." The greatest minds tried and failed. So this ten-year-old boy, Andrew Wiles, sees this problem and it captivates him. He says, "Here was a problem that I, a ten-year-old, could understand… and I knew from that moment that I would never let it go. I had to solve it." Lucas: That’s a heavy vow for a ten-year-old. Most of us are just trying to figure out how to trade lunch snacks. So he grows up and becomes a mathematician, obviously. Christopher: He does. A brilliant one at Princeton. But for decades, he doesn't touch Fermat's Last Theorem. It was seen as a bit of a graveyard for geniuses, a problem for cranks and amateurs. But then, in 1986, he hears a rumor. A breakthrough has happened that connects Fermat's problem to a huge, modern, and very respectable area of mathematics. Suddenly, his childhood dream wasn't a fool's errand anymore. It was the key to a grand, unified theory. Lucas: So he had a legitimate excuse to chase his white whale. What did he do? Christopher: He made a decision that is almost unheard of in modern science. He decided to work in complete and total secrecy. For the next seven years, he told no one what he was doing, not even his closest colleagues. He retreated to his attic study and began his secret calculation. Lucas: Seven years? In total isolation? That’s… intense. Why the secrecy? In science, isn't collaboration everything? You publish, you get feedback, you build on each other's work. Christopher: It is, but this was different. The problem was too famous. The moment he announced he was working on it, the scrutiny would be unbearable. He said, "You can’t really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed." He only confided in his wife, Nada. To the outside world, he just seemed to be fading away professionally, publishing minor papers to keep up appearances. Lucas: Wow. So he’s basically living a double life. The mild-mannered professor by day, and the lone genius wrestling with a 350-year-old demon by night. So after seven years of this, he finally cracks it? Christopher: He thinks he does. In June 1993, he schedules a series of three lectures at Cambridge. The rumors start flying. By the third lecture, the room is packed. People are standing in the aisles, photographers are outside. He fills the blackboard with calculations, and at the very end, he writes down the statement of Fermat's Last Theorem, turns to the audience, and says, with incredible understatement, "I think I'll stop here." Lucas: Chills. That must have been an incredible moment. The room must have erupted. Christopher: It did. Champagne corks were popping. He was on the front page of the New York Times. He had solved the unsolvable. But then… the story takes a turn. A proof that long—200 pages—has to be peer-reviewed. It's sent out to a handful of experts to check every single line. Lucas: And I’m guessing they found a typo. Christopher: It was much, much worse than a typo. A fellow Princeton professor, Nick Katz, was assigned to check a key part of the argument. For months, he and Wiles went back and forth over email about a small point. Wiles was confident it was a minor fix. But one day in September, Wiles looked at the problem again and the floor just fell out from under him. The error wasn't minor. It was a fundamental, gaping hole in the logic. The entire structure of the proof was threatening to collapse. Lucas: Oh, no. So he announces to the whole world he's solved it, and then realizes… it's wrong? That is a public nightmare. What did he do? Christopher: He went back into isolation. But this time it wasn't the joyful, obsessive secrecy of creation. It was a year of what he called "hell." He described it like being in a room where the carpet doesn't fit. You push it down in one corner, and a lump pops up in another. Every time he tried to patch the hole, it created a new problem somewhere else. The world's media, which had celebrated him, was now hounding him, asking if the proof was dead. Lucas: That’s heartbreaking. The pressure must have been immense. This problem has a history of destroying people. The book talks about Évariste Galois, the young French genius who died in a duel at 20, and Yutaka Taniyama, the brilliant Japanese mathematician whose suicide was linked to his work on the very conjecture Wiles was using. This problem has a body count. Christopher: It does. And Wiles was on the verge of admitting defeat. He was ready to publish his work as a noble failure. But his colleague, Richard Taylor, convinced him to give it one last try. And then, on September 19, 1994, almost exactly a year after he found the flaw, he was sitting at his desk, about to give up. He decided to take one last look at why his method was failing. And in that moment, he had a flash of insight. Lucas: The eureka moment! Christopher: The eureka moment. He realized that the very reason the method was failing was the key to making another, older technique he had abandoned years ago work perfectly. The two methods, one flawed and one incomplete, fit together like a lock and key. He said, "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 proof was complete. For real this time.

Lucas: That whole saga is just an incredible human story. But it brings me back to my first question. What was it about this specific problem that drove people mad for centuries? It looks so simple on the page. Christopher: That’s the genius of it. Singh’s book argues that Fermat's Last Theorem is a "perfect" problem for a few reasons. First, its pedigree. It comes directly from Pythagoras, the foundation of mathematics that every schoolkid learns. It feels familiar and fundamental. Lucas: Okay, so it’s accessible. You don't need a Ph.D. to understand the question, just to answer it. Christopher: Precisely. Second, there’s that infuriating marginal note from Fermat. The claim of a "marvelous proof" turned it from a mathematical statement into a treasure hunt. For centuries, mathematicians weren't just trying to prove it; they were trying to rediscover Fermat's lost, elegant proof. They assumed it must be simple, that they were just missing some clever 17th-century trick. Lucas: But Wiles's proof was 200 pages of hyper-modern, complex math. So did Fermat actually have a proof? Christopher: Almost certainly not. Most historians and mathematicians, including Wiles himself, believe Fermat was mistaken. He likely had an idea for a proof that worked for a specific case and wrongly assumed it would generalize. His "marvelous proof" was probably a marvelous error. But the myth was more powerful than the reality. Lucas: The myth fueled the quest. And that quest had some strange consequences. I was fascinated by the story of the Wolfskehl Prize. Christopher: It’s one of the best stories in the book. In the early 20th century, a German industrialist and amateur mathematician named Paul Wolfskehl was heartbroken over a failed love affair and decided to end his life. He was a meticulous man, so he planned everything, setting the date and time for his suicide at midnight. Lucas: Morbidly organized. Christopher: Very. He finished writing his will and farewell letters with a few hours to spare. To pass the time, he went to the library and started reading a paper by the mathematician Ernst Kummer, which was about the failures to prove Fermat's Last Theorem. As he read, he found what he thought was a logical gap in Kummer's argument. He became completely engrossed, trying to fix the flaw. By the time the sun came up, he had proven that Kummer's logic was, in fact, sound. Lucas: Wait, so he got so distracted by a math problem that he forgot to... Christopher: He forgot to die. Mathematics had saved his life. He was so grateful that he rewrote his will, establishing a prize of 100,000 Marks—a fortune at the time—to be awarded to the first person who could prove Fermat's Last Theorem. Lucas: That is unbelievable. So this prize money suddenly makes this obscure problem a public challenge. Christopher: It unleashed a flood. Thousands of amateur proofs poured into the University of Göttingen. They were all wrong, of course. One professor had printed postcards made that just said, "Dear Sir or Madam, your proof of Fermat's Last Theorem has been received and found to be incorrect." But it kept the problem alive in the public imagination. Lucas: So the failures were actually more important than the goal? It seems like every attempt to solve this "useless" problem accidentally created something valuable. Christopher: That’s the deepest insight of the book. The pursuit of Fermat's Last Theorem drove the development of entire new branches of mathematics. Sophie Germain, a brilliant female mathematician in the 1800s who had to use a male pseudonym to even be taken seriously, developed a grand new strategy. Ernst Kummer, in trying to fix a flaw in another proof, invented a whole new class of numbers called "ideal numbers," which revolutionized number theory. The problem was like a whetstone that sharpened the tools of mathematics for three centuries. The journey was truly the destination.

Lucas: So, after all that, after 358 years of struggle, a secret seven-year obsession, a public failure, and a moment of divine insight... what did solving it actually do for the world? If it was useless before, is it still useless now? Christopher: In a practical sense, yes. Your phone won't work better because Fermat's Last Theorem is proven. But in a much more profound sense, the impact was revolutionary. Wiles didn't just solve one problem. To do it, he had to prove a huge part of something called the Taniyama-Shimura conjecture. Lucas: Okay, in English, what does that mean? Christopher: It proposed a radical, beautiful bridge between two completely different and distant worlds of mathematics: the world of elliptic equations and the world of modular forms. Before, they were like separate continents. Proving the conjecture meant showing there was a secret tunnel, a dictionary that could translate between them. It was a massive step towards a "Grand Unified Mathematics," the dream of showing that all the disparate parts of the mathematical universe are secretly connected. Lucas: So solving this one ancient riddle revealed a hidden structure of reality. It wasn't about the answer to xⁿ + yⁿ = zⁿ. It was about discovering the map of the mathematical cosmos. Christopher: Exactly. Wiles himself described the process of research perfectly. He said, "One enters the first room of the mansion and it’s dark. Completely dark. One stumbles around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark." Lucas: That’s a fantastic metaphor for any creative or intellectual struggle. It’s mostly fumbling in the dark, punctuated by these brilliant flashes of light. It makes you wonder what 'useless' problems we're ignoring today that might hold the keys to the next big breakthrough. Christopher: It’s a great question. And it’s a reminder that the most profound journeys are often the ones without a clear, practical destination. After it was all over, Wiles said, "I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream." There's a beauty in that obsession. What's a problem, big or small, that you've been obsessed with? Let us know. We love hearing about those personal quests. Lucas: A beautiful thought to end on. The value of the glorious, useless quest. Christopher: This is Aibrary, signing off.