The Tearful Epiphany of Andrew Wiles

23-10-2026 : গণিত | লেখক: পংকজ জ্যোতি মহন্ত 0 Comments

— The Abel Committee.

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On March 15, 2016, it was announced that Sir Andrew Wiles would be awarded the 2016 Abel Prize “for his stunning proof of Fermat’s Last Theorem (FLT)…, opening a new era in number theory,” a proof he completed in 1994 of a theorem that had remained unproven for over 350 years, despite the efforts of countless brilliant students of mathematics and many renowned mathematicians across the centuries. Established in 2003, the highly prestigious Abel Prize is often regarded as the Nobel Prize of mathematics, and its monetary award is now worth approximately 6.5 crore rupees. Wiles, currently a research professor at the University of Oxford, received the prize on May 24. Following the announcement, Marcus du Sautoy, a renowned Oxford mathematician, popular author, contributor to various BBC programmes promoting science, mathematics and coding, TED speaker, and newly elected Fellow of the Royal Society, wrote in an article: “There are certain moments in history where everyone remembers what they were doing when they heard it happened. The announcement in 1993 that Fermat’s Last Theorem had finally been proven by Andrew Wiles was one such moment for the mathematical community.”

From June 21 to June 23, 1993, Andrew Wiles delivered a series of lectures entitled ‘Modular Forms, Elliptic Curves, and Galois Representations’ at a conference held at the Isaac Newton Institute in Cambridge. Several of the world’s leading number theorists were present. On the final day of the three-day lecture series, after about twenty minutes of speaking and explanation, Wiles wrote Fermat’s Last Theorem on the board as a corollary. The quiet, soft-spoken mathematician then turned towards the audience and humbly announced, “I will stop here.” Although rumors had been swirling during those days of lectures that “Andrew Wiles will prove Fermat’s Last Theorem tomorrow/today,” the moment he wrote the theorem on the board, the hall, filled with over two hundred mathematicians, fell into a pin-drop silence. And the very next moment, the entire hall erupted in thunderous applause. The following day, front-page news broke in The New York Times: “At Last, Shout of ‘Eureka!’ in Age-Old Math Mystery.” Gap Inc., the giant clothing retailer company, reached out with an offer to make Wiles as a model. A group of mathematicians rented out a 1200-seat movie theatre for a public lecture about the theorem, while each ticket was priced at $5, tickets began selling on the black for $25 each. By the end of December, Wiles’s name had appeared on People magazine’s list of the 25 Most Intriguing People of the Year, alongside Michael Jackson, Oprah Winfrey, Tom Hanks, Bill and Hillary Clinton, and others. In short, the announcement had created a global sensation. However, after that lecture, when Wiles submitted his 200-page manuscript to the journal Inventiones Mathematicae for publication, the editor divided it into six parts and assigned each part to two separate reviewers, one of whom was Nicholas Katz. After Katz, along with the other mathematician, spent about two months meticulously examining his assigned portion with Wiles’ collaboration, a flaw was caught towards the end of August. In fact, it was a gap that required inventing entirely new concepts over again to bridge. The mathematics involved was so abstract that even exceptionally gifted number theorists might need two or three months merely to understand it fully. Several months passed, and despite repeated attempts to fix the gap, the prospects of success remained slim. The situation was fraught with tension— would this proof, like so many purported proofs before it, end up bursting like a bubble of jubilation? After all, it is notoriously known as the theorem for which the highest number of erroneous proofs have been announced! Finally, on September 19, 1994, Wiles achieved the ultimate breakthrough and completed the proof.

This theorem, which has deprived countless people of sleep for nearly 350 years, is associated with three thrilling stories:

1. The story of Pierre de Fermat as he proposed the theorem,

2. The story of humanity’s efforts over three and a half centuries to prove it, through which number theory was gradually advanced and elevated,

3. The story of Andrew Wiles.

The latter two stories, beyond the excitement and fascination, also involve tales of profound frustration, superhuman toil, and an intense, haunting passion that constantly churns and torments the mind— a love for the pursuit that one simply cannot abandon, even when trying to give it up.

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Pierre de Fermat was born in the first decade of the seventeenth century into an affluent family in France (died in 1665). He studied law and practiced as a magistrate, receiving early promotions in his professional career. At that time in France, people associated with the legal profession had to remain largely withdrawn from social contact. Consequently, Fermat began pursuing mathematics in his spare time. However, he had virtually no interest in publishing his work and would decline whenever anyone suggested that he do so. He published only one piece of writing, and even that appeared merely as an appendix to a book by a friend. Fermat had several friends who were mathematicians. He jotted down his mathematical work in the margins of books he read or on loose sheets of paper, and would sometimes send it to those friends in the form of letters. These writings were accompanied by extremely brief explanations or, in many cases, no proofs at all. Since this habit often caused difficulties for his peers, some of them would occasionally call him names such as a braggart or a troublemaker out of sheer frustration. Nevertheless, he went on to make incredible contributions to several branches of mathematics. Sir Isaac Newton (1643–1727) acknowledged that much of the foundation for his invention of calculus largely derived from Fermat’s work on tangents to curves at particular points. Blaise Pascal (1623–1662) wrote to Fermat seeking his insights on probability, and the ideas that emerged from their exchange laid the foundations of modern probability theory. René Descartes (1596–1650), widely regarded as the father of coordinate geometry, was also a friend of Fermat. Fermat, too, independently developed coordinate geometry. However, since Fermat’s work remained unpublished during his lifetime, Descartes’s work was published much earlier and consequently became better known. Moreover, Fermat’s work was more advanced. Fermat also made significant advancements in optics. In an article, Simon Singh, the author of several popular books on mathematics and science, including Fermat’s Last Theorem, wrote that Fermat fundamentally changed the very world we live in, because probability theory is used everywhere, from insurance companies to stock markets, while calculus is employed by everyone from architects to NASA. Yet, despite all these monumental contributions, the field to which Fermat contributed his greatest legacy was number theory. Some even regard him as the father of modern number theory.

Fermat had a habit of making an observation or proposing a conjecture and sometimes leaving notes such as, “I can prove this and that, but I have to go feed the cat now,” or “I can solve this particular equation, but I need to wash my hair now,” and so on. After his death, others discovered these scattered notes that he had left here and there. Fermat had done something similar in the margin of a page in an edition of Diophantus’s Arithmetica.

Diophantus (circa 200 AD – 285 AD) is often called the father of algebra. His Arithmetica is a treatise comprising 13 books containing 130 “problems”, though only six of these books have survived to the present day. It was composed around 240–250 AD. In one section of the book, Diophantus discussed various aspects of the Pythagorean theorem, which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of its other two sides. That is, if the hypotenuse of a right-angled triangle has length $z$ , and its other two sides have lengths $x$ and $y$ , then $x^2+y^2=z^2.$ Historical evidence suggests that the Babylonians, Indians, and mathematicians in several other parts of the world knew this property independently long before Pythagoras (c. 570–495 BC). However, Pythagoras is credited with the theorem because historical accounts attribute the first formal proof to him. Today, this theorem can easily be proven in more than a thousand different ways. In that section of Arithmetica, while discussing various aspects of the theorem, Diophantus specifically asked the reader to find non-zero integer solutions for the equation $x^2 + y^2 = z^2.$ This equation is known as the Pythagorean equation. Its general solutions can now be found using nothing more than basic high school algebra. One solution is $(3,4,5)$ , because $3^2+4^2=5^2.$ Similarly, another solution is $(5,12,13)$ , because $5^2+12^2=13^2.$ An infinite number of solutions can be derived from any such solution. For example, starting with $(3,4,5)$ , we obtain

$\quad 3^2+4^2=5^2$

$\Rightarrow a^2(3^2+4^2)=a^2(5^2)$

$\Rightarrow (3a)^2+(4a)^2=(5a)^2.$

Thus, by taking different nonzero integer values of $a$ , we obtain infinitely many solutions such as $(6,8,10)$ ; $(9,12,15)$ ; $(12,16,20)$ ; and so on. Any three integers that satisfy the Pythagorean equation are collectively called a Pythagorean triple.

Of course, back then, the discussion was not expressed in the language we use today. Diophantus framed it as “dividing a square into two squares.” While studying these problems, Fermat thought beyond squares to consider higher powers. In the margin of a page in Arithmetica, he wrote: “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or, in general, any power higher than the second into two like powers.” Alongside this statement, he added the famous, extraordinary remark which destined to echo most loudly throughout the world of mathematics: “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

The original Latin reads:

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”

In English translation:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

The problem can be expressed in modern notation as follows:

The equation $x^n + y^n = z^n$ has no integer solutions for $n > 2$ and $xyz \neq 0.$

Fermat wrote that statement around 1637. Admirers of his work were concerned that if left unpublished, his entire legacy might eventually be lost. His son, Samuel, decided to gather his notes and publish them. In 1670, when Samuel published a new edition of Diophantus’s Arithmetica, incorporating Fermat’s observations, that statement also came to light. By studying Fermat’s notes, mathematicians managed to prove or disprove all his conjectures and theorems within roughly a century—except for this single one. It was the only conjecture that remained unproven. As the last of Fermat’s conjectures and theorems left to be proven, it came to be known as Fermat’s Last Theorem. The equation itself became known as Fermat’s equation. It emerged as a formidable challenge to mathematicians. The question was remarkably simple to understand, yet a century had passed without anyone finding a proof—and all the while, Fermat had written that he possessed one! This made people more intrigued. One century after another passed, until three and a half centuries had gone by.

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Fermat had provided a detailed proof for only one result related to number theory, and that was: “The area of a right-angled triangle cannot be a square number.” That, too, was discovered after his death, written as a note in the margin of a page in Diophantus’s Arithmetica. Fermat believed that this proof would lead to significant advances in number theory. Using this very proof, Fermat’s Last Theorem could be proven for the case $n = 4.$

In 1753, Leonhard Euler (1707–1783) presented a proof for the case $n = 3,$ though a flaw was later found in it. Despite the error, his approach opened an important pathway toward proving the conjecture. Later, mathematicians such as Carl Friedrich Gauss (1777–1855), Johann Dirichlet (1805–1859), and Ernst Kummer (1810–1893) built upon his ideas. After Euler, the next major significant contribution came from Sophie Germain (1776–1831). She studied prime numbers $n$ for which $2n+1$ is also prime. (An example of such a number is 5, since 5 is prime and $2\times5 + 1 = 11$ is also prime.) For these numbers, she showed that if the equation $x^n + y^n = z^n$ has an integer solution, then at least one of $x, y,$ or $z$ must be divisible by $n.$ Consequently, Fermat’s Last Theorem was divided into two cases:

1. None of $x, y,$ or $z$ is divisible by $n$ .

2. Exactly one of $x, y,$ or $z$ is divisible by $n.$

Around 1825, she had proved the first case for those prime numbers.

Sophie Germain’s life story is compelling not only from the perspective of mathematical history but also when viewed through the lens of social history. Her story is sure to stir the minds and hearts of readers. She lived in a terrible era, shrouded in superstition and prejudice! As a woman, she had to endure extreme “intellectual isolation.” In those days, in that society, women who pursued the study of mathematics were not looked upon favorably by society. A book entitled History of Mathematics, which she found in her father’s library, first sparked her interest in mathematics through its account of Archimedes’ death. If a single geometric problem could lead a man to the jaws of death, she wondered, mathematics must be a fascinating subject indeed! She began studying number theory and calculus, and very soon she was delving into the works of Euler and Newton. However, astonished by their daughter’s sudden obsession with mathematics, her parents reprimanded her and tried to pull her away from it. Her father even began taking away her candles, warm clothing, and other comforts! Germain was shy by nature, but she was also exceptionally resolute. She secretly wrapped herself in bedsheets and continued studying. Eventually, her parents’ hearts softened, and her father began supporting her at every stage. Still, because there was no one connected to mathematics in her household, entering the world of mathematicians was thwarted for many years; furthermore, professors did not take her seriously. Meanwhile, a student at a prestigious institute for science and mathematics had left the institution without informing the administration. Consequently, lecture notes, assignments, and other materials continued to be printed for him every week. Germain found a way to collect those materials regularly and began submitting solutions to the problems every week under that student’s name. The evaluator for those assignments was none other than the great mathematician Joseph-Louis Lagrange (1736–1813), and Germain’s solutions were exceptionally original, marked by undeniable brilliance. Naturally, Lagrange could not resist the desire to meet the “student,” and Germain was then compelled to reveal her true identity…. The story of Germain’s life becomes even more inspiring from that point onward. At any rate, Lagrange carried forward some of the subsequent work based on Germain’s contributions to Fermat’s Last Theorem.

The next major breakthrough came from Ernst Kummer. In 1850, he proved Fermat’s Last Theorem for a special class of prime numbers, which he himself named “regular primes.”

Toward the end of Kummer’s life, an unusual incident took place. A young man named Paul Wolfskehl (1856–1906) hailed from a wealthy family. Just as he was preparing to begin his career as a physician, he developed an illness that made him realize he would not be able to practice medicine for very long. Consequently, he decided to abandon medicine and turn to mathematics instead, reasoning that even if he became disabled and had to spend the rest of his life in a wheelchair, he would still be able to continue working on mathematical problems. Wolfskehl had the opportunity to attend Kummer’s lectures, and inspired by the work and enthusiasm of the seventy-three-year-old Kummer, he grew deeply interested in number theory. Naturally, he became acquainted with Fermat’s Last Theorem. In his will, he bequeathed a prize of 100,000 Marks to the first person to prove it—an amount worth several millions in today’s currency. Various sources offer a different tale regarding this incident. According to one, overwhelmed by illness and heartbreak, Wolfskehl had decided to take his own life, planning to shoot himself at midnight. Having made this decision, he sat restlessly in his private library, browsing through various books and waiting for the night to grow late. Just then, his eyes fell upon one of Kummer’s papers on Fermat’s Last Theorem, and a possible flaw in the argument caught his attention. He began examining it carefully. Several hours slipped by while he worked, until he finally realized that the error was in his own reasoning, not Kummer’s. By then, however, the scheduled time for his fatal plan had long passed. Though his attempt to disprove the paper failed, the effort itself and the exquisite mathematical beauty rekindled a fierce passion for life within him. He picked up his existing will and rewrote it, declaring a 100,000 Mark prize for whoever could solve the very “problem” that had swept away his suicidal thoughts and restored his will to live.

After Wolfskehl’s death in 1906, when the prize was announced, Fermat’s Last Theorem became more famous than ever, sparking an even greater rush of people eager to prove it. In the very next year alone, over six hundred proofs were submitted! “Fermat’s Last Theorem” also became a popular subject for limericks, stories, jokes, and satire. At the Eighth Street BMT subway station in New York City, a piece of graffiti even appeared on the wall:

“ $x^n + y^n = z^n$ : No solutions.

I have discovered a truly marvelous proof of this,

but I can’t write it down now, because my train is coming.”

Even celebrated mathematicians like Godfrey Harold Hardy (a name immensely familiar to Indians due to his association with Srinivasa Ramanujan) became involved in such jokes. Hardy was an agnostic and possessed a deep fear of travel. On one occasion, he had to travel by air, so he wrote a note saying, “I have discovered a proof of Fermat’s Last Theorem,” and left it on his desk. Eager to discover the proof, the entire Cambridge community was abuzz with excitement. But when Hardy returned, he simply explained, “I left the note behind to make sure that God would bring me back alive to prove that I was a fraud.”

There were other incidents of a similar kind. The number theorist Edmund Landau had a pre-printed form created to inform senders of the validity of their submitted proofs. It read: “On page ___, from line ___ to line ___, you will find an error.” Martin Gardner, a mathematician and columnist for Scientific American, even had a postcard printed so that he could politely decline whenever readers sent him their proposed proofs for evaluation.

In the period following World War II, computers were used to verify Fermat’s Last Theorem for values of $n$ up to 500. The limit was gradually increased to 1,000 and then 10,000, and reached 25,000 by the 1980s. Building on Kummer’s work and utilizing sophisticated computational techniques of modern computers, mathematicians were able to verify the theorem for values of $n$ up to four million by 1993. However, in the words of Andrew Wiles himself: “Mathematicians are not satisfied with knowing that there are no solutions up to four million or four billion. They really want to know that there are no solutions up to infinity. And for that, we need a proof.”

Amidst all this, an entirely different line of inquiry had begun to develop in 1955. In 1949, a student named Gorō Shimura (1930–2019) had entered the University of Tokyo. In the aftermath of World War II, the professors seemed exhausted and dispirited. Students therefore had to rely on one another for support and inspiration. Shimura formed a collaboration with Yutaka Taniyama (1927–1958). Together, they studied elliptic curves and modular forms. Mathematically, Taniyama was not a particularly meticulous worker—he made many mistakes, but his errors pointed in a fruitful direction, one that ultimately helped lead to the correct answers. Even so, he sometimes faced ridicule because of them. In 1955, at an international symposium, Taniyama presented several problems. From these, Taniyama and Shimura developed the revolutionary idea that elliptic curves and modular forms are equivalent—an idea that later became known as the Taniyama–Shimura Conjecture. It was revolutionary precisely because it proposed an equivalence between two seemingly completely distinct mathematical objects with no obvious connection. At the time, however, no one could have imagined that this conjecture would one day become the master key to the proof of Fermat’s Last Theorem. Tragically, Taniyama ended his own life in 1958, before ever witnessing the momentous impact of his ideas. And the young woman he was engaged to marry, to whom he had promised to remain together for life, followed her beloved’s path just three weeks later.

In 1984, the mathematician Gerhard Frey proposed that if the Taniyama–Shimura conjecture were true, then Fermat’s Last Theorem would follow as a direct consequence. Frey’s key idea was that if Fermat’s Last Theorem were false—that is, if Fermat’s equation had a solution—it would yield a special kind of elliptic curves that could never be modular. On the other hand, according to the Taniyama–Shimura conjecture, all elliptic curves are modular! Just two years later, Ken Ribet successfully proved that Frey’s conjecture was correct!

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Eric Temple Bell (1883–1960) (better known as E. T. Bell), a mathematician, poet, and author of several books on mathematics, including Men of Mathematics, as well as a number of science-fiction novels, famously hailed Fermat as the “Prince of Amateurs.” Bell inspired generations of students to study mathematics and pursue it as a profession through his writings. Among those inspired by it were John Nash, Terence Tao, and Andrew Wiles. At the age of ten, while browsing through the mathematics section of his local library, Wiles stumbled upon another of E. T. Bell’s books, titled The Last Problem. Wiles had loved solving mathematical problems from a very young age. Moreover, whenever he encountered a problem at school, he would return home, construct another problem of his own based on it, and then try to solve that one too. The entire book The Last Problem devoted to just one problem: Fermat’s Last Theorem. What captivated the ten-year-old Wiles was its sheer simplicity—a problem so straightforward that even a child could understand it, yet one that had remained unsolved for three hundred years despite the efforts of many of the greatest mathematicians in human history. Wiles was living in a world three centuries beyond Fermat’s time, in a future that Fermat himself could never have imagined. An idea occurred to him: even in his childhood, Wiles had access to far more mathematical knowledge than Fermat had acquired in his entire lifetime. Inspired by this thought, he set out to find the solution Fermat claimed to have had, dreaming that he would be the one to solve it. But, by the time he reached college, Wiles had come to realize that numerous mathematicians had already worked on the problem throughout the eighteenth and nineteenth centuries. He began studying their methods. By the twentieth century, the methods being applied had been in use for over a hundred years, yet there was no guarantee that they were helping anyone gain a deeper understanding of the core problem. Consequently, working on such a problem meant risking years of fruitless effort, and many advised Wiles to steer clear of it. In 1974, he began his doctoral research at Cambridge under the supervision of Professor John Coates, who offered the exact same advice. Wiles, too, came to recognize the wisdom of this counsel and set the problem aside—yet it never completely left his mind. Wiles received his doctorate in 1980, and in 1981 he was appointed to a position at the Institute for Advanced Study in Princeton. In 1982, he joined Princeton University as a professor.

It was a summer evening in 1986. Wiles was having tea at a friend’s house when, in the middle of their conversation, his friend casually mentioned that Ken Ribet had proven a connection between the Taniyama–Shimura conjecture and Fermat’s Last Theorem. The moment Wiles heard the news, a jolt like an electric shock surged through Wiles’s entire body! From that very instant, the course of his life began to change. To prove Fermat’s Last Theorem, Wiles now had just one task left to accomplish: he had to prove the Taniyama–Shimura conjecture. Since the Taniyama–Shimura conjecture was a problem in modern algebra, any progress toward proving it would be considered immensely valuable in the professional mathematical community, even if a complete proof did not emerge quickly. This was a mainstream problem of the academic realm, meaning that working on it would not invite scorn or skepticism from his peers. Moreover, Wiles was an expert in both elliptic curves and modular forms. His doctoral research under John Coates had been specifically on elliptic curves. Thus, this time, Wiles had no intention of abandoning his pursuit to fulfill his childhood dream. However, there was one major obstacle: no one knew how to proceed toward proving the Taniyama–Shimura conjecture. To achieve it, he would have to invent entirely new methods of his own.

Many believed that proving the Taniyama–Shimura conjecture would take generations. Others even scoffed that everything associated with attempts to prove Fermat’s Last Theorem had always led to nothing, so the Taniyama–Shimura conjecture would not yield anything either! Yet no one could have anticipated that, just seven years later, someone would step onto the stage with a revolutionary proof—such an utterly dramatic proof that it would fundamentally transform the entire scene. For Wiles had made an astounding decision!

As soon as people learned that his work was connected to Fermat’s Last Theorem, it would inevitably attract widespread attention. If such outside attention distracted him, how would it be possible to concentrate fully on a task—purely abstract in nature—that might take years to complete? Wiles came up with a clever strategy. At that very time, he had a major piece of research ready for publication. He immediately decided not to publish it all at once; instead, he chose to divide it into sections and release them at intervals of six or seven months. This would reassure his colleagues that he was carrying on with his research as usual. Wiles was so extraordinarily creative and highly productive that he had already earned a reputation as one of the finest mathematicians of his generation. Yet as the years went by and his appearances at conferences, seminars, and other academic gatherings dwindled to virtually zero, many of his peers began to suspect that his brilliance had faded. Some even thought that Wiles was finished! Meanwhile, in complete secrecy, he relentlessly pursued the proof of the Taniyama–Shimura conjecture—a mission that no one could even suspect in the slightest. Only his wife, Nada, knew about it, having been told just two nights after their wedding. Nada herself was exceptionally brilliant, holding a doctorate in molecular biology from Princeton. She had heard of Fermat’s Last Theorem before, of course, but at that time she had little idea of the extent to which it had captivated and challenged the mathematical world. Wiles had decided to immerse his life completely between nothing but the problem and his family.

Whatever new ideas Wiles had to discover, he later remarked that where they came from was itself a mystery! Day and night, morning and evening, that very problem revolved incessantly in his mind. Whenever he faced a daunting impasse—a moment when he had absolutely no idea how to proceed or what new framework he needed to invent next, he would usually go for a walk by the nearby lake. During those walks, he felt a sense of relaxation, but in reality, it was precisely then that he allowed his subconscious mind to work most actively. In Wiles’s own words, the entire endeavor was like a journey through a dark, unfamiliar mansion. First, you enter a room and, in the darkness, feel your way around the furniture, gradually discovering what the room is like and what lies inside… until suddenly, after six months, you stumble upon the light switch. The moment you turn it on, everything becomes brilliantly illuminated, and you finally understand where you have actually arrived. After that, you must enter the next dark room, and the same process of exploration begins all over again. Wiles knew that he was heading in the right direction, but he also realized he might never reach his ultimate destination—for the method required for the next stage might take a hundred years to invent. As he put it, “So even if I was on the right track, I could be living in the wrong century.”

Toward the end, Wiles discussed his proof separately with two colleagues, Nick Katz (the reviewer mentioned earlier) and Peter Sarnak. Both were exceptional experts in number theory, and Wiles trusted them implicitly to keep the matter strictly confidential. They honored that trust completely. Sarnak later recalled that upon hearing the news, he was flabbergasted, feeling utterly overwhelmed with emotion and excitement—so restless that he found it nearly impossible to sleep that night.

Finally, after seven years, Wiles became convinced that he had completed the proof of the theorem. Around that time, John Coates was organizing a conference at the Isaac Newton Institute in Cambridge. Wiles considered Cambridge—the very place where he had dreamed of proving the theorem as a young boy—to be the ideal setting to present his proof to the world. Barry Mazur, a mathematician attending the conference, who would later serve as the editor of Wiles’s manuscript and divide it into six parts for distribution among the referees, remarked three years later, “I’d never seen a lecture series in mathematics like that before. What was unique about those lectures were the glorious ideas, how many new ideas were presented, and the constancy of his dramatic build-up that was suspenseful until the end.”

After the flaw in the proof came to light, many urged Wiles to release a draft of the manuscript so that others could try to help resolve it. But Wiles refused. It was time for him to withdraw back into intense focus and devote himself single-mindedly to the work once again—yet it was also a deeply humbling and painfully embarrassing ordeal. Only the referees and a few of Wiles’s most trusted friends knew the exact nature of the flaw. Sarnak advised Wiles that it would be wise to bring in collaborator with whom he felt comfortable working closely. Wiles contacted his former doctoral student, Richard Taylor, who had by then established himself as a leading number theorist at Cambridge. An agonizing year slipped by, yet their efforts seemed to be getting them nowhere; the proof remained incomplete. His colleagues were at a loss for words, unsure of how to even approach him. Sometimes, one would tell another, “I saw Wiles this morning.” The other would ask, “Was he smiling?” To which the reply might be, “Yes… he smiled, but he didn’t seem happy….” Eventually, Wiles was on the verge of giving up and accepting defeat, deciding that he would have to abandon the effort. But on September 19, 1994, he achieved the ultimate breakthrough and fully resolved the flaw. The proof was subsequently published in the May 1995 issue of Annals of Mathematics as two research papers: Wiles’s ‘Modular Elliptic Curves and Fermat’s Last Theorem’ and a joint companion paper with Taylor, titled ‘Ring-Theoretic Properties of Certain Hecke Algebras.’

The question naturally arises: could Fermat have arrived at this very same proof 350 years ago? No, that is impossible. For this reason, some still hold out hope that a simpler, more elementary proof might yet be discovered someday. On the other hand, it is also quite possible that Fermat made a subtle error in his reasoning that he himself never realized—yet his conjecture turned out to be true all along!

Wiles did not need to prove the entire Taniyama–Shimura conjecture; he proved it only for a particular class of elliptic curves—the precise portion required to settle Fermat’s Last Theorem. Following the approach developed by Wiles, a team of mathematicians including Taylor went on to prove the conjecture in its full generality (now known as the modularity theorem) about four years later.

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Before writing his book Fermat’s Last Theorem, Simon Singh directed a 50-minute documentary of the same name in 1996. The highly acclaimed documentary won several awards, including the BAFTA Award for Best Documentary. It was later retitled The Proof.

The documentary captures Wiles’s thirty-year-long, all-consuming obsession with the problem and the eight years of extraordinary dedication and painstaking effort he devoted to solving it in grueling isolation. In it, whenever he recalls that pivotal day of September 19, he repeatedly becomes overwhelmed with emotion. Having already resolved to abandon the effort, he had decided to look at the work one last time—desperately trying to pinpoint what the exact issue was and why approach was not working… when all of a sudden, beyond belief, he experienced an incredible revelation. As he tries to describe that feeling, Wiles’s eyes well up with tears. He says that it was the most important moment of his working life, and once again, emotion takes over. Unable to continue as his feelings overwhelm him, he falls completely silent for a long moment. Eventually, attempting to speak again, he explains that what he had discovered was so indescribably beautiful, extraordinarily simple, elegant, and perfectly harmonious that he sat there in utter disbelief, staring at it, transfixed, for twenty minutes. Later that day, he went to his department and, upon returning, looked at it again—yes, it was still working. That night, he slept on it. The following day, he checked it once more, and yes, his proof was complete. Satisfied at last, he went downstairs to see his wife, Nada, “I’ve got it, I think I’ve got it, I’ve found it….” The documentary’s opening minute also shows a few moments from this deeply emotional experience, where Wiles tries to say, “Nothing I ever do again will…”, but he is unable to continue and falls silent once more. Finally, offering an embarrassed, modest smile, he whispers, “I’m sorry,” and turns his face away from the camera….

That elusive insight, held back despite endless painstaking cultivation, suddenly pierced through the dense veil of mystery to illuminate everything with astonishing clarity. In that single instant, Wiles ascended from the bottomless abyss of despair to the absolute summit of realization—a moment of total oneness with the act of creation itself, a connection bound to slip away just moments later. Was Wiles moved to tears remembering that moment because he realized he would never again experience such a rare, profound moment of human ingenuity—a moment spent witnessing pure, unblemished mathematical beauty?

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Now, someone might wonder: “Fermat’s Last Theorem was finally proven true after so much monumental effort, but what do we actually gain from it? We now know that a cube cannot be split into two cubes, nor can two cubes be added together to form a single cube, and that the same holds true for all higher powers. But what practical benefit does any of this offer us?”

Isn’t this question fundamentally the same as asking, “What do we gain when humans explore the the creatures that living beneath the ocean’s surface?” Or, “What did we gain from people climbing Mount Everest, despite facing such immense perils to reach its summit?”

This is, above all, a story of human curiosity and the pursuit of truth.

However, in the case of Fermat’s Last Theorem, it would be a mistake to dismiss its consequences as merely abstract. Although the theorem itself has no direct, literal application today, the efforts devoted over 350 years toward its proof enriched a whole branch of mathematics, and many of the ideas developed along the way are now being used in our everyday lives. For instance, modern algebra, which emerged from this mathematical progress, is used to secure online transactions, digital data transmission, security systems, and much more. Whenever we withdraw money using an ATM card or purchase something with a debit card, we unknowingly benefit from these mathematical developments. Perhaps only around 200 people worldwide fully understand the underlying mathematics, yet its benefits reach all of us. Even during the early stages of this mathematical development, when the seeds were first sown half a century ago, nobody could have imagined that it would eventually prove useful in such applications. Moreover, it is entirely possible that one day the theorem itself may find direct, practical applications as well. Wiles, too, once remarked in another context that the true measure of a good mathematical problem lies not just in the problem itself, but in the new mathematics it generates along the course of finding its solution.

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An Assamese mathematician also earned his doctorate under John Coates, the Cambridge professor and Fellow of the Royal Society under whom Andrew Wiles had pursued his doctoral research. His name is Anupam Saikia. A few years ago, in an interview for Gonit Sora, I asked him, “After Fermat’s Last Theorem was proven, Andrew Wiles was asked in an interview that since the theorem has no practical applications and is merely an abstract mathematical question, why do people strive so hard for such a proof? How would you answer the same question? And has the proof led to the discovery of any new directions or developments that are being used today?”

He replied:

“In my opinion, it is a good thing that humanity does not spend all its time thinking solely about practical applications. Many great minds pursue truth for its own sake, whether in mathematics or in philosophy. They are driven by an insatiable thirst for knowledge and remain unsatisfied until they uncover the truth. Abstract concepts are absolutely essential for expanding the horizons of human knowledge. Moreover, there are numerous examples of ideas that began in the purely abstract realm, but later found practical applications. Fermat’s theorem has not yet found any direct practical application, but it has played a profoundly important role in the advancement of mathematics.”