Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem

Just before dawn on June 23, 1993, Professor John Conway approached the darkened mathematics building on the Princeton University campus.  He unlocked the front door and quickly walked up to his office.  For weeks preceding his colleague Andrew Wiles' departure for England, persistent but unspecific rumors had been circulating in the world's mathematical community.  Conway was expecting something important to happen.  Exactly what it was, he had no idea.  He turned on his computer and sat down to stare at the screen.  At 5 53 AM, a terse e-mail message flashed from across the Atlantic "Wiles proves F.L.T."
Cambridge, England, June 1993

Late in June of 1993, Professor Andrew Wiles flew to England.  He was returning to Cambridge University, where he had been a graduate student twenty years earlier.  Wiles' former doctoral thesis adviser at Cambridge, Professor John Coates, was organizing a conference on Iwasawa Theory--the particular area within number theory in which Andrew Wiles did his dissertation and about which he knew a great deal.  Coates had asked his former student if he would mind giving a short, one-hour talk at the conference on a topic of his choice.  To his great surprise and that of the other conference organizers, the shy Wiles--previously reluctant to speak in public--responded by asking if he could be given three hours of presentation.

The 40-year-old Wiles looked the typical mathematician when he arrived in Cambridge: white dress shirt with sleeves rolled up carelessly, thick horn-rimmed glasses, unruly strands of thinning light hair.  Born in Cambridge, his return was a very special kind of homecoming--it was the realization of a childhood dream.  In pursuit of this dream, Andrew Wiles had spent the last seven years of his life a virtual prisoner in his own attic.  But he hoped that soon the sacrifice, the years of struggle and the long hours of solitude would end.  Soon he might be able to spend more time with his wife and daughters, of whom he had seen so little for seven years.  He had often failed to show up for lunch with his family, missed afternoon tea, barely made it to dinner.  But now the accolades would be his alone.

The Sir Isaac Newton Institute for Mathematical Sciences at Cambridge had only recently opened by the time Professor Wiles arrived to deliver his three hour-long lectures.  The Institute is spacious, set in scenic surroundings at some distance from the University of Cambridge.  Wide areas outside the lecture halls are furnished with plush, comfortable chairs, designed to help facilitate the informal exchange of ideas among scholars and scientists, and to promote learning and knowledge.

Although he knew most of the other mathematicians who came to the specialized conference from around the world, Wiles kept to himself.  When colleagues became curious about the length of his scheduled presentation, Wiles would only say they should come to his lectures and find out for themselves.  Such secretiveness was unusual, even for a mathematician.  While they often work alone trying to prove theorems and are generally not known to be the world's most gregarious people, mathematicians usually share research results with each other.  Mathematical results are freely circulated by their authors in the form of research preprints.  These preprints bring their authors outside comments that help them improve the papers before they are published.  But Wiles didn't hand out preprints and didn't discuss his work.  The title of Wiles' talks was "Modular Forms, Elliptic Curves, and Galois Representations," but the name gave no hint where the lectures would lead, and even experts in his field could not guess.  The rumors intensified as time went on.

On the first day, Wiles rewarded the 20 or so mathematicians who came to his lecture with a powerful and unexpected mathematical result--and there were still two more lectures to go.  What was coming?  It became clear to everyone that Wiles' lectures were the place to be, and the suspense grew as expectant mathematicians flocked to the lectures.

On the second day, Wiles' presentation intensified.  He had brought with him over 200 pages of formulas and derivations, original thoughts stated as new theorems with their lengthy, abstract proofs.  The room was now filled to capacity.  Everyone listened intently.  Where would it lead?  Wiles gave no hint.  He dispassionately continued writing on the blackboard and when he was done for the day, he quickly disappeared.

The next day, Wednesday, June 23, 1993, was his last talk.  As he neared the lecture hall, Wiles found it necessary to push his way in.  People stood outside blocking the entrance and the room was overflowing.  Many carried cameras.  As Wiles again wrote seemingly endless formulas and theorems on the board, the tension increased.  "There was only one possible climax, only one possible end to Wiles' presentation," Professor Ken Ribet of the University of California at Berkeley later told me.  Wiles was finishing the last few lines of his proof of an enigmatic and complicated conjecture in mathematics, the Shimura-Taniyama Conjecture.  Then suddenly he added one final line, a restatement of a centuries-old equation, one which Ken Ribet had proved seven years earlier would be a consequence of the conjecture.  "And this proves Fermat's Last Theorem," he said, almost offhandedly.  "I think I'll stop here."

There was a moment of stunned silence in the room.  Then the audience erupted in spontaneous applause.  Cameras flashed as everyone stood up to congratulate a beaming Wiles.  Within minutes, electronic mail flashed and faxes rolled out of machines around the world.  The most celebrated mathematical problem of all time appeared to have been solved.

"What was so unexpected was that the next day we were deluged by the world press," recalled Professor John Coates, who organized the conference without having the slightest idea that it would become the launching ground for one of the greatest mathematical achievements.  Headlines in the world's newspapers hailed the unexpected breakthrough.  "At Last, Shout of 'Eureka!' In Age-Old Math Mystery" announced the front page of the New York Times on June 24, 1993.  The Washington Post called Wiles in a major article "The Math Dragon-Slayer," and news stories everywhere described the person who apparently solved the most persistent problem in all of mathematics, one that had defied resolution for over 350 years.  Overnight, the quiet and very private Andrew Wiles became a household name.
Pierre de Fermat

Pierre de Fermat was a seventeenth-century French jurist who was also an amateur mathematician.  But while he was technically an "amateur" since he had a day job as a jurist, the leading historian of mathematics E. T. Bell, writing in the early part of the twentieth century, aptly called Fermat the "Prince of Amateurs."  Bell believed Fermat to have achieved more important mathematical results than most "professional" mathematicians of his day.  Bell argued that Fermat was the most prolific mathematician of the seventeenth century, a century that witnessed the work of some of the greatest mathematical brains of all time.

One of Fermat's most stunning achievements was to develop the main ideas of calculus, which he did thirteen years before the birth of Sir Isaac Newton.  Newton and his contemporary Gottfried Wilhelm von Leibniz are jointly credited in the popular tradition with having conceived the mathematical theory of motion, acceleration, forces, orbits, and other applied mathematical concepts of continuous change we call calculus.

Fermat was fascinated with the mathematical works of ancient Greece.  Possibly he was led to his conception of calculus ideas by the work of the classical Greek mathematicians Archimedes and Eudoxus, who lived in the third and fourth centuries B.C., respectively.  Fermat studied the works of the ancients--which were translated into Latin in his day--in every spare moment.  He had a full-time job as an important jurist, but his hobby--his passion--was to try to generalize the work of the ancients and to find new beauty in their longburied discoveries.  "I have found a great number of exceedingly beautiful theorems," he once said.  These theorems he would jot down in the margins of the translated copies of ancient books he possessed.

Fermat was the son of a leather merchant, Dominique Fermat, who was Second Consul in the town of Beaumont-de-Lomagne, and of Claire de Long, the daughter of a family of parliamentary judges.  The young Fermat was born in August 1601 (baptized August 20 in Beaumont-de-Lomagne), and was raised by his parents to be a magistrate.  He went to school in Toulouse, and was installed in the same city as Commissioner of Requests at the age of thirty.  He married Louise Long, his mother's cousin, that same year, 1631.  Pierre and Louise had three sons and two daughters.  One of their sons, Clement Samuel, became his father's scientific executor and published his father's works after his death.  In fact, it is the book containing Fermat's work, published by his son, that has come down to us and from which we know his famous Last Theorem.