All elementary math - average math internet school - great mathematicians - wiles. Fermat's Theorem: The Story of Andrew Wiles Proof Andrew Wiles Brief Biography

Andrew Wiles is a professor of mathematics at Princeton University, he proved Fermat's Last Theorem, over which more than one generation of scientists struggled for hundreds of years.

30 years on one task

Wiles first learned about Fermat's Last Theorem when he was ten years old. He stopped by on his way home from school to the library and became interested in reading the book "The Last Task" by Eric Temple Bell. Perhaps without knowing it, but from that moment on he devoted his life to finding proof, despite the fact that it was something that eluded the best minds on the planet for three centuries.

Wiles learned about Fermat's last theorem when he was ten years old.

He found it 30 years later after another scientist, Ken Ribet, proved the connection between the theorem of Japanese mathematicians Taniyama and Shimura and Fermat's Last Theorem. Unlike skeptical colleagues, Wiles immediately understood - this is it, and seven years later he put an end to the proof.

The process of proof itself turned out to be very dramatic: Wiles completed his work in 1993, but right during a public speech he found a significant "gap" in his reasoning. It took two months to find an error in the calculations (the error was hidden among 130 printed pages of solving the equation). Then, for a year and a half, hard work was carried out to correct the error. The entire scientific community of the Earth was at a loss. Wiles completed his work on September 19, 1994, and immediately presented it to the public.

frightening fame

Most of all, Andrew was afraid of fame and publicity. For a very long time he refused to appear on television. It is believed that John Lynch was able to convince him. He assured Wiles that he could inspire a new generation of mathematicians and show the power of mathematics to the public.

Andrew Wiles turned down TV appearances for a long time

A little later, a grateful society began to reward Andrew with awards. So on June 27, 1997, Wiles received the Wolfskel Prize, which was approximately $50,000, much less than Wolfskel had intended to keep a century earlier, but hyperinflation has reduced the amount.

Unfortunately, the mathematical equivalent of the Nobel Prize, the Fields Prize, simply did not go to Wiles due to the fact that it is awarded to mathematicians under the age of forty. Instead, he received a special silver plate at the Fields Medal ceremony in honor of his important achievement. Wiles has also won the prestigious Wolf Prize, the King Faisal Prize and many other international awards.

Opinions of colleagues

The reaction of one of the most famous contemporary Russian mathematicians, Academician V. I. Arnold, to the proof is "actively skeptical":

This is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its very nature, cannot generate the development of mathematics, since it is "binary", that is, the formulation of the problem requires an answer only to the question "yes or no".

However, mathematical work recent years V. I. Arnold himself was largely devoted to variations on very close number-theoretic topics. It is possible that Wiles, paradoxically, became an indirect cause of this activity.

real dream

When Andrew is asked how he managed to sit in four walls for more than 7 years, doing one task, Wiles tells how he dreamed during his work thatthe time will come when mathematics courses in universities, and even in schools, will be adjusted to his method of proving the theorem. He wanted the very proof of Fermat's Last Theorem to become not only a model mathematical problem, but also a methodological model for teaching mathematics. Wiles imagined that on her example it would be possible to study all the main branches of mathematics and physics.

4 ladies without whom there would be no proof

Andrew is married and has three daughters, two of whom were born "in the seven-year process of the first version of the proof."

Wiles himself believes that without his family he would not have succeeded.

During these years, only Nada, Andrew's wife, knew that he alone stormed the most impregnable and most famous peak of mathematics. It is to them, Nadia, Claire, Kate and Olivia, that Wiles's famous final article "Modular Elliptic Curves and Fermat's Last Theorem" is dedicated in the central mathematical journal Annals of Mathematics, which publishes the most important mathematical works. However, Wiles himself does not at all deny that without his family he would not have succeeded.

Mathematician of Princeton University, head of its department of mathematics, member of the scientific council.

The highlight of his career was the 1994 proof of Fermat's Last Theorem. In 2016, he was awarded the Abel Prize for this proof.

Fermat's Last Theorem

Wiles' work is fundamental, but the method is applicable only to elliptic curves over rational numbers. Perhaps there is a more general proof that elliptic curves are modular.

Reflection in culture

Wiles' work on Fermat's Last Theorem was featured in the musical Fermat's Great Tango by Lessner and Rosenbloom.

Wiles and his work are mentioned in the Star Trek: Deep Space Nine episode "Facets".

Awards

Andrew Wiles is the recipient of many international prizes in mathematics, including:

• Wolfskel Prize (1996)
• US National Academy of Sciences Mathematics Award (1996)
• Ostrovsky Prize (1996)
• Wolf Prize in Mathematics (1996)
• Silver plate from the International Mathematical Union ()
• King Faisal International Prize (1998)
• (1999)
• Knight Commander of the Order of the British Empire (2000)

see also

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Notes

Logic and philosophy Maths Music visual arts Rafael Moneo (1993) · Claes Oldenburg (1995) · Thorsten Andersson (1997) · Herzog and de Meuron (1999) · Giuseppe Penone (2001) · Susan Rothenberg (2003) · SANAA / Kazuyo Seijima + Ryuyo Nishizawa (2005) · Mona Hatoum (2008) · and or external references, but the sources of individual claims remain unclear due to lack of footnotes. Statements not , may be questioned and removed. You can improve the article by adding more precise references to the sources.

Excerpt characterizing Wiles, Andrew John

- They are! Dear fathers! .. By God, they are. Four, mounted! .. - she shouted.
Gerasim and the janitor let go of Makar Alekseich, and in the quiet corridor they distinctly heard the knock of several hands on the front door.

Pierre, who decided with himself that he did not need to reveal either his rank or knowledge until the fulfillment of his intention French, stood in the half-open doors of the corridor, intending to hide at once, as soon as the French entered. But the French entered, and Pierre still did not leave the door: irresistible curiosity held him back.
There were two of them. One is an officer, tall, brave and handsome man, the other is obviously a soldier or orderly, a squat, thin, tanned man with sunken cheeks and a dull expression on his face. The officer, leaning on a stick and limping, walked ahead. Having taken a few steps, the officer, as if deciding with himself that this apartment was good, stopped, turned back to the soldiers standing in the doorway and in a loud commanding voice shouted to them to bring in the horses. Having finished this business, the officer with a gallant gesture, raising his elbow high, straightened his mustache and touched his hat with his hand.
Bonjour la compagnie! [Respect to the entire company!] – he said cheerfully, smiling and looking around. Nobody answered.
– Vous etes le bourgeois? [Are you the boss?] – the officer turned to Gerasim.
Gerasim looked inquiringly at the officer in fright.
“Quartire, quartire, logement,” said the officer, looking down at him with a condescending and good-natured smile. little man. – Les Francais sont de bons enfants. Que diable! Voyons! Ne nous fachons pas, mon vieux, [Apartments, apartments… The French are good guys. Damn it, let's not quarrel, grandfather.] - he added, patting the frightened and silent Gerasim on the shoulder.
– A ca! Dites donc, on ne parle donc pas francais dans cette boutique? [Well, doesn't anyone speak French here, too?] he added, looking around and meeting Pierre's eyes. Pierre moved away from the door.
The officer again turned to Gerasim. He demanded that Gerasim show him the rooms in the house.
"No master - don't understand... my yours..." said Gerasim, trying to make his words clearer by speaking them backwards.
The French officer, smiling, spread his hands in front of Gerasim's nose, making it feel that he did not understand him either, and, limping, went to the door where Pierre was standing. Pierre wanted to move away in order to hide from him, but at that very moment he saw Makar Alekseich leaning out of the kitchen door opening with a pistol in his hands. With the cunning of a madman, Makar Alekseevich looked at the Frenchman and, raising his pistol, took aim.
- Aboard!!! - the drunk shouted, pressing the trigger of the pistol. The French officer turned around at the cry, and at the same moment Pierre rushed at the drunk. While Pierre grabbed and raised the pistol, Makar Alekseich finally hit the trigger with his finger, and a shot rang out that deafened and doused everyone with powder smoke. The Frenchman turned pale and rushed back to the door.
Having forgotten his intention not to reveal his knowledge of the French language, Pierre, snatching the pistol and throwing it away, ran up to the officer and spoke to him in French.
- Vous n "etes pas blesse? [Are you injured?] - he said.
“Je crois que non,” the officer answered, feeling himself, “mais je l "ai manque belle cette fois ci,” he added, pointing to the chipped plaster in the wall. “Quel est cet homme? [It seems not ... but this once it was close. Who is this man?] - looking sternly at Pierre, the officer said.
- Ah, je suis vraiment au desespoir de ce qui vient d "arriver, [Ah, I really am in despair over what happened,] - Pierre said quickly, completely forgetting his role. - C" est un fou, un malheureux qui ne savait pas ce qu "il faisait. [This is an unfortunate madman who did not know what he was doing.]
The officer went up to Makar Alekseevich and seized him by the collar.
Makar Alekseich, with parted lips, as if falling asleep, swayed, leaning against the wall.
“Brigand, tu me la payeras,” said the Frenchman, withdrawing his hand.
– Nous autres nous sommes clements apres la victoire: mais nous ne pardonnons pas aux traitres, [Robber, you will pay me for this. Our brother is merciful after the victory, but we do not forgive the traitors,] he added with gloomy solemnity in his face and with a beautiful energetic gesture.
Pierre continued to persuade the officer in French not to exact from this drunken, insane man. The Frenchman listened in silence, without changing his gloomy look, and suddenly turned to Pierre with a smile. He looked at him silently for a few seconds. His handsome face took on a tragically tender expression, and he held out his hand.
- Vous m "avez sauve la vie! Vous etes Francais, [You saved my life. You are a Frenchman,]" he said. For a Frenchman, this conclusion was undeniable. Only a Frenchman could do a great deed, and saving his life, m r Ramball "I capitaine du 13 me leger [Monsieur Rambal, captain of the 13th light regiment] was, without a doubt, the greatest deed.
But no matter how undoubted this conclusion and the officer’s conviction based on it, Pierre considered it necessary to disappoint him.
“Je suis Russe, [I am Russian],” Pierre said quickly.
- Ti ti ti, a d "autres, [tell it to others] - said the Frenchman, waving his finger in front of his nose and smiling. - Tout a l "heure vous allez me conter tout ca," he said. – Charme de rencontrer un compatriote. Eh bien! qu "allons nous faire de cet homme? [Now you will tell me all this. It is very nice to meet a compatriot. Well! what should we do with this man?] - he added, addressing Pierre, already as his brother. If only Pierre was not a Frenchman, having once received this highest name in the world, he could not renounce it, said the expression on the face and tone of the French officer. To the last question, Pierre once again explained who Makar Alekseich was, explained that just before their arrival this a drunken, insane man dragged away a loaded pistol, which they did not have time to take away from him, and asked that his deed be left without punishment.

Sir Andrew John Wiles(Eng. Sir Andrew John Wiles, born April 11, 1953, Cambridge, UK, Knight Commander of the Order of the British Empire since 2000) is an English and American mathematician, professor of mathematics at Princeton University, head of his department of mathematics, member of the scientific council of the Clay Institute of Mathematics.

He received his bachelor's degree in 1974 from Merton College, Oxford University. He began his scientific career in the summer of 1975 at Clare College, Cambridge University, where he received his doctorate. From 1977 to 1980, Wiles served as an Associate Fellow at Clare College and Associate Professor at Harvard University. He worked on the arithmetic of elliptic curves with complex multiplication using the methods of Iwasawa theory. In 1982, Wiles moved from the UK to the US.

The highlight of his career was the proof of Fermat's Last Theorem in 1994. In 2016, he was awarded the Abel Prize for this proof.

Fermat's Last Theorem

Fermat's Last Theorem states that there are no natural solutions to the equation an + bn = cn for natural numbers n > 2.

Andrew Wiles learned about Fermat's Last Theorem at the age of ten. Then he made an attempt to prove it using methods from a school textbook. Later, he began to study the work of mathematicians who tried to prove this theorem. After entering college, Andrew gave up trying to prove Fermat's Last Theorem and took up the study of elliptic curves under John Coates.

He began working on Fermat's theorem in the summer of 1986, immediately after Ken Ribet proved that Fermat's theorem follows from the Taniyama-Shimura conjecture in the case of semistable elliptic curves.

Wiles' work is fundamental, but the method is applicable only to elliptic curves over rational numbers. Perhaps there is a more general proof that elliptic curves are modular.

Reflection in culture

Wiles' work on Fermat's Last Theorem was featured in the musical Fermat's Great Tango by Lessner and Rosenbloom.

Wiles and his work are mentioned in the Star Trek: Deep Space Nine episode "Facets".

Awards

Andrew Wiles is the recipient of many international prizes in mathematics, including:

• Whitehead Award (1988)
• Shock Award (1995)
• Fermat Prize (1995)
• Wolfskel Prize (1996)
• US National Academy of Sciences Mathematics Award (1996)
• Ostrovsky Prize (1996)
• Royal Medal (1996)
• Wolf Prize in Mathematics (1996)
• Cole Award (1997)
• MacArthur Fellowship (1997)
• Silver plate from the International Mathematical Union (1998)
• King Faisal International Prize (1998)
• Clay Institute of Mathematics Award (1999)
• Knight Commander of the Order of the British Empire (2000)
• Shao Award (2005)
• Abel Prize (2016)

In 2000 he made a plenary report at the European Mathematical Congress.

Andrew John Wiles(b. April 11, 1953, Cambridge, UK, Knight Commander of the Order of the British Empire since 2000) - an outstanding English and American mathematician, professor and head of the Department of Mathematics at Princeton University, member of the Scientific Council of the Clay Institute of Mathematics.
He received his bachelor's degree in 1974 from Merton College, Oxford University. He began his scientific career in the summer of 1975 under the guidance of Professor John Coates at Clare College, Cambridge University, where he received his doctorate. From 1977 to 1980, Wiles served as an Associate Fellow at Clare College and Associate Professor at Harvard University. Together with John Coates, he worked on elliptic curve arithmetic with complex multiplication using the methods of Iwasawa theory. In 1982, Wiles moved from the UK to the US.
One of the highlights of his career was the proof Fermat's Last Theorem in 1993 and the discovery of a technical method that allowed him to complete the proof with the help of his former graduate student, R. Taylor, in 1994. He began working on Fermat's theorem in the summer of 1986 after Ken Ribet proved the conjecture about the connection between semistable elliptic curves (a special case of the Taniyama-Shimura theorem) and Fermat's theorem. The basic idea of ​​such a connection belongs to the German mathematician Gerhard Frei. Fermat's Last Theorem states that there are no natural solutions to the Diophantine equation x n + y n = z n for natural n > 2.
Andrew Wiles learned about Fermat's Last Theorem at the age of ten. Then he made an attempt to prove it using methods from a school textbook; Naturally, he didn't succeed. Later, he began to study the work of mathematicians who tried to prove this theorem. After entering college, Andrew gave up trying to prove Fermat's Last Theorem and took up the study of elliptic curves under John Coates.
In the 1950s and 1960s, a connection between elliptic curves and modular forms was suggested by the Japanese mathematician Shimura, who built on ideas expressed by another Japanese mathematician, Taniyama. In Western scientific circles, this hypothesis was known thanks to the work of André Weil, who, as a result of careful analysis of it, found partial evidence in favor of it. Because of this, the conjecture is often referred to as the Shimura–Taniyama–Weil theorem. The theorem says that every elliptic curve over an algebraic number field is automorphic. In particular, every elliptic curve over rational numbers must be modular (certain analytic functions of a complex variable are modular). The last property was fully proved in 1998 by Christoph Broglie, Brian Conrad, Fred Diamond and Richard Taylor, who tested some degenerate cases, supplementing the most general case considered by Wiles in 1995. Certainly, Wiles' work is fundamental. However, his method is very special and only works for elliptic curves over rational numbers, while the Taniyama-Shimura conjecture covers elliptic curves over any algebraic number field. Based on this, it is reasonable to assume that there is a more general and more elegant proof of the modularity of elliptic curves.
Andrew Wiles is the recipient of many international prizes in mathematics, including:
Shock Award (1995).
Cole Award (1996).
National Academy of Sciences Award in Mathematics from the American Mathematical Society (1996).
Ostrovsky Prize (1996).
Royal Medal (1996).
Wolf Prize in Mathematics (1996).
Wolfskel Prize (1997).
MacArthur Fellowship (1997).
Silver plate from the International Mathematical Union (1998).
King Faisal Prize (1998).
Clay Mathematical Institute Award (1999).
Knight Commander of the Order of the British Empire (2000).
Shaw Award (2005).

In the last twentieth century, an event occurred on a scale that has never been equaled in mathematics in its entire history. On September 19, 1994, a theorem formulated by Pierre de Fermat (1601-1665) over 350 years ago in 1637 was proved. It is also known as "Fermat's last theorem" or as "Fermat's great theorem" because there is also the so-called "Fermat's little theorem". It was proved by 41-year-old, up to this point in the mathematical community nothing particularly unremarkable, and by mathematical standards already middle-aged, Princeton University professor Andrew Wiles.

It is surprising that not only our ordinary Russian inhabitants, but also many people who are interested in science, including even a considerable number of scientists in Russia who use mathematics in one way or another, do not really know about this event. This is shown by the incessant "sensational" reports about the "elementary proofs" of Fermat's theorem in Russian popular newspapers and on television. The latest evidence was covered with such information power, as if Wiles's proof, which had passed the most authoritative examination and received the widest fame all over the world, did not exist. The reaction of the Russian mathematical community to this front-page news in the situation of a rigorous proof obtained long ago turned out to be amazingly sluggish. Our aim is to sketch out the fascinating and dramatic story of Wiles' proof in the context of the fairy tale of Fermat's greatest theorem, and to talk a little about the proof itself. Here, we are primarily interested in the question of the possibility of an accessible presentation of Wiles' proof, which, of course, most mathematicians in the world know about, but only very, very few of them can talk about understanding this proof.

So, let's remember Fermat's famous theorem. Most of us have heard of her in one way or another since we were in school. This theorem is related to a very significant equation. This is perhaps the simplest meaningful equation that can be written using three unknowns and one more strictly positive integer parameter. Here it is:

Fermat's Last Theorem states that for values ​​of the parameter (the degree of the equation) greater than two, there are no integer solutions to this equation (except, of course, the solution when all these variables are equal to zero at the same time).

The attractive power of this Fermat's theorem for the general public is obvious: there is no other mathematical statement that has such simplicity of formulation, the apparent accessibility of the proof, as well as the attractiveness of its "status" in the eyes of society.

Before Wiles, an additional incentive for fermatists (as the people who maniacally attacked Fermat’s problem were called) was the German Wolfskell’s proof prize, established almost a hundred years ago, though small compared to Nobel Prize- it managed to depreciate during the First World War.

In addition, the probable elementality of the proof was always attracted, since Fermat himself "proved it" by writing on the margins of the translation of Diophantus' Arithmetic: "I have found a truly wonderful proof for this, but the margins here are too narrow to accommodate it."

That is why it is appropriate here to give an assessment of the relevance of popularizing Wiles' proof of Fermat's problem, which belongs to the famous American mathematician R. Murty (we quote from the soon-to-be-published translation of the book "Introduction to Modern Number Theory" by Yu. Manin and A. Panchishkin):

Fermat's Last Theorem holds a special place in the history of civilization. With its external simplicity, it has always attracted both amateurs and professionals ... Everything looks as if it was conceived by some higher mind, which over the centuries has developed various directions of thought only to then reunite them into one exciting fusion to solve the Big Fermat's theorems. No person can claim to be an expert on all the ideas used in this "wonderful" proof. In an era of general specialization, when each of us knows "more and more about less and less", it is absolutely necessary to have an overview of this masterpiece ... "

Let's start with a short historical digression, largely inspired by Simon Singh's fascinating book Fermat's Last Theorem. Around the insidious theorem, alluring with its apparent simplicity, serious passions have always boiled. The history of her proof is full of drama, mysticism and even direct victims. Perhaps the most iconic victim is Yutaka Taniyama (1927-1958). It was this young talented Japanese mathematician, who in life was distinguished by great extravagance, created the basis for Wiles's attack in 1955. Based on his ideas, Goro Shimura and André Weil a few years later (60-67 years) finally formulated the famous conjecture, proving a significant part of which, Wiles obtained Fermat's theorem as a corollary. The mysticism of the story of the death of the non-trivial Yutaka is connected with his stormy temperament: he hanged himself at the age of thirty-one on the basis of unhappy love.

The whole long history of the enigmatic theorem was accompanied by constant announcements of its proof, starting with Fermat himself. Constant errors in an endless stream of proofs comprehended not only amateur mathematicians, but also professional mathematicians. This has led to the fact that the term "fermatist", applied to Fermat's theorem provers, has become a household word. The constant intrigue with its proof sometimes led to amusing incidents. So, when a gap was discovered in the first version of Wiles' already widely publicized proof, a snide inscription appeared at one of the New York subway stations: "I found a truly wonderful proof of Fermat's Last Theorem, but my train came and I do not have time to write it down."

Andrew Wiles, born in England in 1953, studied mathematics at Cambridge; in graduate school was with Professor John Coates. Under his guidance, Andrew comprehended the theory of the Japanese mathematician Iwasawa, which is on the border of classical number theory and modern algebraic geometry. Such a fusion of seemingly distant mathematical disciplines was called arithmetic algebraic geometry. Andrew challenged Fermat's problem, relying precisely on this synthetic theory, which is difficult even for many professional mathematicians.

After graduating from graduate school, Wiles received a position at Princeton University, where he still works. He is married and has three daughters, two of whom were born "in the seven-year process of the first version of the proof." During these years, only Nada, Andrew's wife, knew that he alone stormed the most impregnable and most famous peak of mathematics. It is to them, Nadia, Claire, Kate and Olivia, that Wiles's famous final article "Modular Elliptic Curves and Fermat's Last Theorem" is dedicated in the central mathematical journal Annals of Mathematics, which publishes the most important mathematical works.

The events around the proof unfolded quite dramatically. This exciting scenario could be called "fermatist-professional mathematician."

Indeed, Andrew dreamed of proving Fermat's theorem since youthful years. But unlike the vast majority of fermatists, it was clear to him that for this he needed to master entire layers of the most complex mathematics. Moving towards his goal, Andrew graduated from the Faculty of Mathematics of the famous University of Cambridge and began to specialize in modern number theory, which is at the junction with algebraic geometry.

The idea of ​​assaulting the shining peak is quite simple and fundamental - the best possible ammunition and careful development of the route.

As a powerful tool for achieving the goal, Wiles himself develops the already familiar Iwasawa theory, which has deep historical roots. This theory generalized Kummer's theory - historically the first serious mathematical theory to storm Fermat's problem, which appeared back in the 19th century. In turn, the roots of Kummer's theory lie in the famous theory of the legendary and brilliant romantic revolutionary Evariste Galois, who died at the age of twenty-one in a duel in defense of the honor of a girl (pay attention, remembering the story with Taniyama, to the fatal role of beautiful ladies in the history of mathematics) .

Wiles is completely immersed in the proof, even stopping participation in scientific conferences. And as a result of a seven-year seclusion from the mathematical community in Princeton, in May 1993, Andrew puts an end to his text - it's done.

It was at this time that a great occasion turned up to notify the scientific world of his discovery - already in June a conference was to be held in his native Cambridge on exactly the right topic. Three lectures at the Cambridge Institute of Isaac Newton excite not only the mathematical world, but also the general public. At the end of the third lecture, on June 23, 1993, Wiles announces the proof of Fermat's Last Theorem. The proof is saturated with a whole bunch of new ideas, such as a new approach to the Taniyama-Shimura-Weil conjecture, a far advanced Iwasawa theory, a new "deformation control theory" of Galois representations. The mathematical community is looking forward to the verification of the text of the proof by experts in arithmetic algebraic geometry.

This is where the dramatic twist comes in. Wiles himself, in the process of communicating with reviewers, discovers a gap in his proof. The crack was given by the mechanism of "deformation control" invented by him - the supporting structure of the proof.

The gap is discovered a couple of months later by Wiles' line-by-line explanation of his proof to a colleague in his Princeton department, Nick Katz. Nick Katz, having been in friendly relations with Andrew, recommends to him cooperation with a promising young English mathematician Richard Taylor.

Another year of hard work passes, connected with the study of an additional tool for attacking an intractable problem - the so-called Euler systems, independently discovered in the 80s by our compatriot Viktor Kolyvagin (already working at New York University for a long time) and Thain.

And here is a new challenge. The unfinished, but still very impressive result of Wiles's work, he reports to the International Congress of Mathematicians in Zurich at the end of August 1994. Wiles fights with all his might. Literally before the report, according to eyewitnesses, he is still feverishly writing something, trying to improve the situation with the “sagging” evidence as much as possible.

After this intriguing audience of the largest mathematicians of the world, Wiles's report, the mathematical community “exhales joyfully” and applauds sympathetically: nothing, the guy, with whomever he happens to, but he advanced science, showing that it is possible to successfully advance in solving such an impregnable hypothesis, which no one has ever done before. didn't even think about doing it. Another fermatist, Andrew Wiles, could not take away the innermost dream of many mathematicians about proving Fermat's theorem.

It is natural to imagine the state of Wiles at that time. Even the support and benevolent attitude of colleagues in the shop could not compensate for his state of psychological devastation.

And so, just one month later, when, as Wiles writes in the introduction to his final proof in the Annals, "I decided to take a last look at the Euler systems in an attempt to revive this argument for proof," it happened. Wiles had a flash of insight on September 19, 1994. It was on this day that the gap in the proof was closed.

Then things moved at a rapid pace. Already established cooperation with Richard Taylor in the study of the Euler systems of Kolyvagin and Thain made it possible to finalize the proof in the form of two large papers already in October.

Their publication, which occupied the entire issue of the Annals of Mathematics, followed already in November 1994. All this caused a new powerful information surge. The story of Wiles's proof received an enthusiastic press in the United States, a film was made and books were published about the author of a fantastic breakthrough in mathematics. In one evaluation of his own work, Wiles noted that he had invented the mathematics of the future.

(I wonder if this is true? We only note that with all this information flurry, there was a sharp contrast to the almost zero information resonance in Russia, which continues to this day).

Let's ask ourselves a question - what is the "inner kitchen" of obtaining outstanding results? After all, it is interesting to know how a scientist organizes his work, what he focuses on in it, how he determines the priorities of his activity. What can be said in this sense about Andrew Wiles? And it suddenly turns out that modern era active scientific communication and collaborative style of work, Wiles had his own way of working on superproblems.

Wiles went to his fantastic result on the basis of intensive, continuous, many years of individual work. The organization of its activities, speaking in official language, was extremely unscheduled. This was categorically not an activity within the framework of a specific grant, for which it is necessary to regularly report and again plan to receive certain results by a certain date each time.

Such activities outside of society, not using direct scientific communication with colleagues, even at conferences, seemed contrary to all the canons of the work of a modern scientist.

But it was individual work that made it possible to go beyond the already established standard concepts and methods. This style of work, closed in form and at the same time free in essence, made it possible to invent new powerful methods and obtain results of a new level.

The problem facing Wiles (the Taniyama-Shimura-Weil conjecture) was not even among the nearest peaks that modern mathematics could conquer in those years. At the same time, none of the experts denied its great importance, and nominally it was in the "mainstream" of modern mathematics.

Thus, Wiles' activities were of a pronounced non-systemic nature and the result was achieved thanks to the strongest motivation, talent, creative freedom, will, more than favorable material conditions for working at Princeton and, most importantly, mutual understanding in the family.

Wiles's proof, which appeared like a bolt from the blue, became a kind of test for the international mathematical community. The reaction of even the most progressive part of this community as a whole turned out to be, oddly enough, rather neutral. After the emotions and enthusiasm of the first time after the appearance of the landmark evidence subsided, everyone calmly continued their business. Experts in arithmetic algebraic geometry slowly studied the "powerful proof" in their narrow circle, while the rest plowed their mathematical paths, diverging, as before, farther and farther from each other.

Let's try to understand this situation, which has both objective and subjective reasons. The objective factors of non-perception, oddly enough, have their roots in the organizational structure of modern scientific activity. This activity is like a skating rink going down a slope with tremendous momentum: its own school, its established priorities, its own sources of funding, and so on. All this is good from the point of view of an established system of reporting to the grantor, but it makes it difficult to raise your head and look around: what is really important and relevant for science and society, and not for the next portion of the grant?

Then - again - I do not want to get out of my cozy mink, where everything is so familiar, and climb into another, completely unfamiliar hole. It is not known what to expect there. Moreover, it is obviously clear that they don’t give money for the invasion.

It is quite natural that none of the bureaucratic structures that organize science in different countries, including Russia, did not draw conclusions not only from the phenomenon of the proof of Andrew Wiles, but also from the similar phenomenon of the sensational proof of Grigory Perelman of another, also famous mathematical problem.

The subjective factors of the neutrality of the reaction of the mathematical world to the "millennium event" lie in quite prosaic reasons. The proof is indeed extraordinarily complicated and lengthy. To the layman in arithmetic algebraic geometry, it seems to consist of a layering of the terminology and constructions of the most abstract mathematical disciplines. It seems that the author did not at all aim at being understood by as many interested mathematicians as possible.

This methodological complexity, unfortunately, is present as an inevitable cost of the great proofs of recent times (for example, the analysis of Grigory Perelman's recent proof of the Poincaré conjecture continues to this day).

The complexity of perception is further enhanced by the fact that arithmetic algebraic geometry is a very exotic subfield of mathematics, causing difficulties even for professional mathematicians. The matter was also aggravated by the extraordinary syntheticity of Wiles's proof, which used a variety of modern tools created by a large number of mathematicians in the most recent years.

But it must be taken into account that Wiles was not faced with the methodical task of explanation - he was constructing a new method. It was the synthesis of Wiles' own brilliant ideas and a conglomeration of the latest results from various mathematical fields that worked in the method. And it was such a powerful design that rammed an impregnable problem. The proof was not accidental. The fact of its crystallization fully corresponded to both the logic of the development of science and the logic of cognition. The task of explaining such a super-proof seems to be absolutely independent, a very difficult, although very promising problem.

You can test public opinion yourself. Try asking mathematicians you know about Wiles' proof: Who got it? Who understood at least the basic ideas? Who wants to understand? Who felt that this is the new mathematics? The answers to these questions seem to be rhetorical. And it is unlikely that you will meet many who want to break through the palisade of technical terms and master new concepts and methods in order to solve just one very exotic equation. And why for the sake of this task it is necessary to study all this?!

Let me give you a funny example. A couple of years ago, the famous French mathematician, Fields laureate, Pierre Deligne, a prominent specialist in algebraic geometry and number theory, when asked by the author about the meaning of one of the key objects of Wiles' proof - the so-called "ring of deformations" - after half an hour of thought, he said that he was not completely understands the meaning of this object. Ten years have passed since the proof.

Now you can reproduce the reaction of Russian mathematicians. The main reaction is its practically complete absence. This is mainly due to Wiles' "heavy" and "unaccustomed" mathematics.

For example, in classical number theory you won't find such long proofs as Wiles's. As number theorists put it, "the proof must be a page" (Wyles's proof, in collaboration with Taylor, is 120 pages long in the journal version).

It is also impossible to exclude the factor of fear for the unprofessionalism of your assessment: in reacting, you take responsibility for assessing the evidence. And how to do it when you do not know this mathematics?

Characteristic is the position taken by direct specialists in number theory: "... and awe, and burning interest, and caution in the face of one of the greatest mysteries in the history of mathematics" (from the preface to Paulo Ribenboim's book "Fermat's Last Theorem for Amateurs" - the only one available today day to source directly on Wiles' proof for the general reader.

The reaction of one of the most famous contemporary Russian mathematicians, Academician V.I. Arnold on the proof is “actively skeptical”: this is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its very nature, cannot generate the development of mathematics, since it is "binary", that is, the formulation of the problem requires an answer only to the question "yes or no". At the same time, the mathematical works of recent years by V.I. Arnold's works turned out to be largely devoted to variations on very close number-theoretic topics. It is possible that Wiles, paradoxically, became an indirect cause of this activity.

At the Mekhmat of Moscow State University, nevertheless, proof enthusiasts appear. The remarkable mathematician and popularizer Yu.P. Solovyov (who died untimely) initiates the translation of E. Knapp's book on elliptic curves with the necessary material on the Taniyama–Shimura–Weil conjecture. Alexey Panchishkin, who is now working in France, in 2001 reads lectures at the Mekhmat, which formed the basis of the corresponding part of his work with Yu.I. Manin of the excellent book mentioned above on modern number theory (published in Russian translation by Sergei Gorchinsky with editing by Alexei Parshin in 2007).

It is somewhat surprising that at the Moscow Steklov Institute of Mathematics, the center of the Russian mathematical world, Wiles' proof was not studied at seminars, but was studied only by individual specialized experts. Moreover, the proof of the already complete Taniyama-Shimura-Weil conjecture was not understood (Wyles proved only a part of it, sufficient for proving Fermat's theorem). This proof was given in 2000 by a whole team of foreign mathematicians, including Richard Taylor, Wiles's co-author on the final stage of the proof of Fermat's theorem.

Also, there were no public statements and, moreover, no discussions on the part of well-known Russian mathematicians about Wiles' proof. A rather sharp discussion is known between the Russian V. Arnold (“a skeptic of the method of proof”) and the American S. Leng (“an enthusiast of the method of proof”), however, its traces are lost in Western publications. In the Russian central mathematical press, since the publication of Wiles' proof, there have been no publications on the subject of the proof. Perhaps the only publication on this topic was the translation of an article by the Canadian mathematician Henry Darmon, even a still incomplete version of the proof in the Mathematical Advances in 1995 (it's funny that the full proof has already been published).

Against this "sleepy" mathematical background, despite the highly abstract nature of Wiles's proof, some intrepid theoretical physicists have included it in the area of ​​their potential interest and began to study it, hoping sooner or later to find applications of Wiles's mathematics. This cannot but rejoice, if only because this mathematics has been practically in self-isolation all these years.

Nevertheless, the problem of adapting the proof, which greatly aggravates its applied potential, has remained and remains very relevant. To date, the original, highly specialized text of Wiles' article and the joint article by Wiles and Taylor has already been adapted, though only for a fairly narrow circle of professional mathematicians. This was done in the mentioned book by Yu. Manin and A. Panchishkin. They succeeded in smoothing over a certain artificiality of the original proof. In addition, the American mathematician Serge Leng, a fierce promoter of Wiles' proof (unfortunately passed away in September 2005), included some of the most important constructions of the proof in the third edition of his now classic university textbook Algebra.

As an example of the artificiality of the original proof, we note that one of the most striking features that gives this impression is the special role of individual prime numbers, such as 2, 3, 5, 11, 17, as well as individual natural numbers, such as 15, 30 and 60. Among other things, it is quite obvious that the proof is not geometric in the most usual sense. It does not contain natural geometric images that could be attached to for a better understanding of the text. The super-powerful "terminologized" abstract algebra and "advanced" number theory purely psychologically hit the perception of the proof of even a qualified reader-mathematician.

One can only wonder why, in such a situation, the experts of the proof, including Wiles himself, do not “polish” him, do not promote and popularize an obvious “mathematical hit” even in the native mathematical community.

So, in short, today the fact of Wiles's proof is simply the fact of the proof of Fermat's theorem with the status of the first correct proof and the "some super-powerful mathematics" used in it.

The well-known Russian mathematician of the middle of the last century, the former dean of the Mekhmat, V.V. Golubev:

“... according to the witty remark of F. Klein, many departments of mathematics are similar to those exhibitions latest models weapons that exist at firms that manufacture weapons; with all the wit invested by the inventors, it often happens that when the real war, these innovations turn out to be unsuitable for one reason or another ... The modern teaching of mathematics presents exactly the same picture; students are given very perfect and powerful means of mathematical research ... but further students cannot bear any idea of ​​where and how these powerful and ingenious methods can be applied in solving the main task of all science: in understanding the world around us and in influencing him the creative will of man. At one time, A.P. Chekhov said that if in the first act of the play a gun is hanging on the stage, then it is necessary that at least in the third act it should be fired. This observation is fully applicable to the teaching of mathematics: if any theory is presented to students, then it is necessary to show sooner or later what applications can be made from this theory, primarily in the field of mechanics, physics or technology and in other areas.

Continuing this analogy, we can say that Wiles's proof is extremely favorable material for studying a huge layer of modern fundamental mathematics. Here students can be shown how the problem of classical number theory is closely related to such areas of pure mathematics as modern algebraic number theory, modern Galois theory, p-adic mathematics, arithmetic algebraic geometry, commutative and non-commutative algebra.

It would be fair if Wiles's confidence that the mathematics he invented - mathematics of a new level was confirmed. And I really don’t want this really very beautiful and synthetic mathematics to suffer the fate of an “unfired gun”.

And yet, let us now ask ourselves the question: is it possible to describe Wiles's proof in sufficiently accessible terms for a wide interested audience?

From the point of view of specialists, this is an absolute utopia. But let's still try, guided by the simple consideration that Fermat's theorem is a statement about just integer points of our usual three-dimensional Euclidean space.

We will sequentially substitute points with integer coordinates into Fermat's equation.

Wiles finds an optimal mechanism for recalculating integer points and testing them for satisfaction of the equation of Fermat's theorem (after introducing the necessary definitions, such a recalculation will just correspond to the so-called "modularity property of elliptic curves over the field of rational numbers", described by the Taniyama-Shimura-Weyl conjecture").

The recalculation mechanism is optimized with the help of a remarkable discovery by the German mathematician Gerhard Frey, who connected the potential solution of Fermat's equation with an arbitrary exponent to another, completely different equation. This new equation is given by a special curve (called the Frey elliptic curve). This Frey curve is given by a very simple equation:

The surprise of Frey's idea was the transition from the number-theoretic nature of the problem to its "hidden" geometric aspect. Namely: Frey compared to any solution of Fermat's equation, that is, to numbers satisfying the relation

the above curve. Now it remains to show that such curves do not exist for . In this case, Fermat's Last Theorem would follow from here. It was this strategy that was chosen by Wiles in 1986, when he began his enchanting assault.

Frey's invention at the time of Wiles's "start" was quite fresh (85th year) and also echoed the relatively recent approach of the French mathematician Hellegouarch (70s), who proposed using elliptic curves to find solutions to Diophantine equations, i.e. equations similar to Fermat's equation.

Let's now try to look at the Frey curve from a different point of view, namely, as a tool for recalculating integer points in Euclidean space. In other words, our Frey curve will play the role of a formula that determines the algorithm for such a recalculation.

In this context, it can be said that Wiles invents tools (special algebraic constructions) to control this recalculation. Strictly speaking, this subtle instrumentation of Wiles constitutes the central core and the main complexity of the proof. It is in the manufacture of these tools that Wiles's main sophisticated algebraic discoveries arise, which are so difficult to perceive.

But still, the most unexpected effect of the proof, perhaps, is the sufficiency of using only one "Freev" curve, which is represented by a completely simple, almost "school" dependence. Surprisingly, the use of only one such curve is sufficient to test all points of the three-dimensional Euclidean space with integer coordinates for satisfaction of their relation of Fermat's Last Theorem with an arbitrary exponent.

In other words, the use of only one curve (albeit one that has a specific form), which is understandable even to an ordinary high school student, turns out to be tantamount to building an algorithm (program) for the sequential recalculation of integer points in ordinary three-dimensional space. And not just a recalculation, but a recalculation with simultaneous testing of the whole point for “its satisfaction” with the Fermat equation.

It is here that the phantom of Pierre de Fermat himself arises, since in such a recalculation what is usually called "Ferma't descent", or Fermat's reduction (or "method of infinite descent") comes to life.

In this context, it immediately becomes clear why Fermat himself could not prove his theorem for objective reasons, although at the same time he could well “see” the geometric idea of ​​its proof.

The fact is that the recalculation takes place under the control of mathematical tools that have no analogues not only in the distant past, but also unknown before Wiles even in modern mathematics.

The most important thing here is that these tools are "minimal", ie. they cannot be simplified. Although in itself this "minimalism" is very difficult. And it was Wiles's realization of this non-trivial "minimalness" that became the decisive final step of the proof. This was exactly the same "flash" on September 19, 1994.

Some problem that causes dissatisfaction still remains here - in Wiles this minimal construction is not explicitly described. Therefore, those interested in Fermat's problem still have interesting work to do - a clear interpretation of this "minimality" is needed.

It is possible that this is where the geometry of the “algebraized” proof should be hidden. It is possible that Fermat himself felt exactly this geometry when he made the famous entry in the narrow margins of his treatise: "I found a truly remarkable proof ...".

Now let's go directly to the virtual experiment and try to "dig into" the thoughts of the mathematician-lawyer Pierre de Fermat.

The geometric image of the so-called Fermat's little theorem can be represented as a circle rolling "without slipping" along a straight line and "winding" integer points around itself. The equation of Fermat's little theorem in this interpretation also acquires a physical meaning - the meaning of the law of conservation of such motion in one-dimensional discrete time.

We can try to transfer these geometric and physical images to the situation when the dimension of the problem (the number of variables in the equation) increases and the equation of Fermat's little theorem turns into the equation of Fermat's big theorem. Namely: let us assume that the geometry of Fermat's Last Theorem is represented by a sphere rolling on a plane and "winding" on itself whole points on this plane. It is important that this rolling should not be arbitrary, but "periodic" (mathematicians also say "cyclotomic"). Rolling periodicity means that the linear and angular velocity vectors of a sphere rolling in the most general way after a certain fixed time (period) are repeated in magnitude and direction. Such a periodicity is similar to the periodicity of the linear velocity of a circle rolling along a straight line, modeling the “small” Fermat equation.

Accordingly, Fermat's "large" equation acquires the meaning of the law of conservation of the above motion of the sphere already in two-dimensional discrete time. Let us now take the diagonal of this two-dimensional time (it is in this step that the whole difficulty lies!). This extremely tricky and turns out to be the only diagonal is the equation of Fermat's Last Theorem when the exponent of the equation is exactly two.

It is important to note that in a one-dimensional situation - the situation of Fermat's Little Theorem - such a diagonal does not need to be found, since time is one-dimensional and there is no reason to take a diagonal. Therefore, the degree of the variable in the equation of Fermat's little theorem can be arbitrary.

So, rather unexpectedly, we get a bridge to the "physicalization" of Fermat's last theorem, that is, to the appearance of its physical meaning. How can one not remember that Fermat was also no stranger to physics.

By the way, the experience of physics also shows that the conservation laws of mechanical systems of the above type are quadratic in the physical variables of the problem. And finally, all this is quite consistent with the quadratic structure of the laws of conservation of energy in Newtonian mechanics, known from the school.

From the point of view of the above "physical" interpretation of Fermat's Last Theorem, the "minimal" property corresponds to the minimal degree of the conservation law (this is two). And the reduction of Fermat and Wiles corresponds to the reduction of the laws of conservation of recalculation of points to the law of the simplest form. This simplest (minimum complexity) recalculation, both geometrically and algebraically, is represented by the rolling of the sphere on the plane, since the sphere and the plane are “minimal”, as we understand it, two-dimensional geometric objects.

The whole complexity, which at first glance is absent, here lies in the fact that an accurate description of such a seemingly “simple” movement of the sphere is not at all easy. The point is that the "periodic" rolling of the sphere "absorbs" a bunch of so-called "hidden" symmetries of our three-dimensional space. These hidden symmetries are due to non-trivial combinations (compositions) of the linear and angular motion of the sphere - see Fig.1.

It is precisely for the exact description of these hidden symmetries, geometrically encoded by such a tricky rolling of the sphere (points with integer coordinates "sit" at the nodes of the drawn lattice), that Wiles's algebraic constructions are required.

In the geometric interpretation shown in Fig. 1, the linear movement of the center of the sphere “counts” integer points on the plane, and its angular (or rotational) movement provides the spatial (or vertical) component of the recalculation. The rotational motion of the sphere is not immediately possible to "see" in the arbitrary rolling of the sphere on the plane. It is the rotational motion that corresponds to the hidden symmetries of the Euclidean space mentioned above.

The Frey curve introduced above just “encodes” the most aesthetically beautiful recalculation of integer points in space, reminiscent of moving along a spiral staircase. Indeed, if we follow the curve swept by some point of the sphere in one period, we will find that our marked point will sweep the curve shown in Fig. 2, resembling a "double spatial sinusoid" - a spatial analogue of the graph. This beautiful curve can be interpreted as a graph of the "minimum" Frey curve. This is the graph of our testing recalculation.

Having connected some associative perception of this picture, to our surprise we will find that the surface bounded by our curve is strikingly similar to the surface of the DNA molecule - the "corner brick" of biology! It is perhaps no coincidence that the terminology of DNA-encoding constructs from Wiles' proof is used in Singh's book Fermat's Last Theorem.

We emphasize once again that the decisive moment of our interpretation is the fact that the analogue of the conservation law for Fermat's Little Theorem (its degree can be arbitrarily large) is the equation of Fermat's Last Theorem precisely in the case of . It is this effect of "minimality of the degree of the law of conservation of rolling of a sphere on a plane" that corresponds to the statement of Fermat's Great Theorem.

It is possible that Fermat himself saw or felt these geometric and physical images, but at the same time could not assume that they are so difficult to describe from a mathematical point of view. Moreover, he could not assume that to describe such a non-trivial, but still sufficiently transparent geometry, it would take another three hundred and fifty years of work by the mathematical community.

Now let's build a bridge to modern physics. The geometric image of Wiles's argument proposed here is very close to the geometry of modern physics trying to get to the enigma of the nature of gravity - quantum general relativity. To confirm this, at first glance unexpected, interaction of Fermat's Last Theorem and "Big Physics", let's imagine that the rolling sphere is massive and "presses through" the plane under it. The interpretation of this "punching" in Fig. 3 strikingly resembles the well-known geometric interpretation of Einstein's general theory of relativity, which describes precisely the "geometry of gravity."

And if we also take into account the present discretization of our picture, embodied by a discrete integer lattice on a plane, then we are completely observing “quantum gravity” with our own eyes!

It is on this major "unifying" physical and mathematical note that we will finish our "cavalry" attempt to give a visual interpretation of Wiles' "super-abstract" proof.

Now, perhaps, it should be emphasized that in any case, whatever the correct proof of Fermat's theorem, it must necessarily use the constructions and logic of Wiles' proof in one way or another. It is simply not possible to get around all this because of the mentioned "minimality property" of Wiles' mathematical tools used for the proof. In our "geometro-dynamical" interpretation of this proof, this "minimality property" provides the "minimum necessary conditions" for the correct (i.e., "converging") construction of the testing algorithm.

On the one hand, this is a huge disappointment for amateur fermatists (unless, of course, they find out about it; as they say, “the less you know, the better you sleep”). On the other hand, the natural "irreducibility" of Wiles' proof formally makes life easier for professional mathematicians - they may not read periodically appearing "elementary" proofs from amateur mathematicians, referring to the lack of correspondence with Wiles's proof.

The general conclusion is that both of them need to “strain themselves” and understand this “savage” proof, comprehending, in essence, “all mathematics”.

What else is important not to miss when summing up this unique story that we have witnessed? The strength of Wiles' proof is that it is not just formal logical reasoning, but is a broad and powerful method. This creation is not a separate tool for proving one single result, but an excellent set of well-chosen tools that allows you to "split" a wide variety of problems. It is also of fundamental importance that when we look down from the height of the skyscraper of Wiles' proof, we see all the previous mathematics. The pathos lies in the fact that it will not be a "patchwork", but a panoramic vision. All this speaks not only of the scientific, but also of the methodological continuity of this truly magical proof. It remains “just nothing” - only to understand it and learn how to apply it.

I wonder what our contemporary hero Wiles is doing today? There is no special news about Andrew. He, of course, received various awards and prizes, including the very famous German Wolfskel Prize, which depreciated during the first civil war. For all the time that has passed since the triumph of the proof of Fermat's problem until today, I managed to notice only one, albeit as always large, article in the same Annals (co-authored with Skinner). Maybe Andrew is hiding again in anticipation of a new mathematical breakthrough, for example, the so-called "abc" hypothesis - recently formulated (by Masser and Osterle in 1986) and considered the most important problem in number theory today (this is the "problem of the century" in the words of Serge Leng ).

Much more information about Wiles' co-author on the final part of the proof, Richard Taylor. He was one of four authors of the proof of the complete Taniyama-Shmura-Weil conjecture and was a serious contender for the Fields Medal at the 2002 Mathematical Congress in China. However, he did not receive it (at that time only two mathematicians received it - the Russian mathematician from Princeton Vladimir Voevodsky "for the theory of motives" and the Frenchman Laurent Laforgue "for an important part of the Langlands program"). Taylor published during this time a considerable number of remarkable works. And just recently, Richard achieved a new great success - he proved a very famous conjecture - the Tate-Saito conjecture, also related to arithmetic algebraic geometry and generalizing the results of German. 19th century mathematician G. Frobenius and 20th century Russian mathematician N. Chebotarev.

Let's finally fantasize a little. Perhaps the time will come when mathematics courses in universities, and even in schools, will be adjusted to the methods of Wiles's proof. This means that Fermat's Last Theorem will become not only a model mathematical problem, but also a methodological model for teaching mathematics. On its example, it will be possible to study, in fact, all the main branches of mathematics. Moreover, future physics, and perhaps even biology and economics, will be based on this mathematical apparatus. But what if?

It seems that the first steps in this direction have already been taken. This is evidenced, for example, by the fact that the American mathematician Serge Leng included in the third edition of his classic manual on algebra the main constructions of Wiles' proof. Go even further Russian Yuri Manin and Aleksey Panchishkin in the mentioned new edition of their "Modern Number Theory", setting out in detail the proof itself in the context of modern mathematics.

And now how not to exclaim: Fermat's great theorem is "dead" - long live the Wiles method!