Fermat's Last Theorem: Wiles and Perelman's proof, formulas, calculation rules and complete proof of the theorem. The sensation around the Farm theorem turned out to be a misunderstanding Sir Andrew's scientific journey

In the last twentieth century, an event occurred that has never been equal in scale in mathematics in its entire history. On September 19, 1994, a theorem formulated by Pierre de Fermat (1601-1665) more than 350 years ago in 1637 was proven. It is also known as "Fermat's last theorem" or "Fermat's last theorem" because there is also the so-called "Fermat's little theorem". It was proved by 41-year-old Princeton University professor Andrew Wiles, who up to this point had been unremarkable in the mathematical community and, by mathematical standards, no longer young.

It is surprising that not only our ordinary Russian inhabitants, but also many people 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 continuous “sensational” reports about “elementary proofs” of Fermat’s theorem in Russian popular newspapers and on television. The latest evidence was covered with such informational power, as if Wiles’ evidence, which had undergone the most authoritative examination and became widely known throughout the world, did not exist. The reaction of the Russian mathematical community to this front-page news in the context of a rigorous proof obtained long ago was surprisingly sluggish. Our goal is to sketch the fascinating and dramatic history of Wiles's proof in the context of the enchanting history of Fermat's great theorem itself, and to talk a little about its 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 about it in one way or another since 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.

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 a given equation (except, of course, the solution when all these variables are equal to zero at the same time).

The attractive power of 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 the Fermatists (as people who maniacally attacked Fermat’s problem were called) was the German Wolfskehl’s prize for proof, established almost a hundred years ago, although small compared to the Nobel Prize - it managed to depreciate during the First World War.

In addition, the probable elementary nature of the proof has always attracted attention, since Fermat himself “proved it” by writing in the margins of the translation of Diophantus’ Arithmetic: “I have found a truly wonderful proof of this, but the margins here are too narrow to contain 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-released translation of the book by Yu. Manin and A. Panchishkin “Introduction to Modern Number Theory”):

“Fermat’s Last Theorem occupies a special place in the history of civilization. With its outward simplicity, it has always attracted both amateurs and professionals... Everything looks as if it was conceived by some higher mind, which over the centuries developed various lines of thought only to then reunite them into one exciting fusion to solve the Great Fermat's theorems. No one person can claim to be an expert on all the ideas used in this “miracle” proof. In an era of universal 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 brief historical excursion, mainly inspired by Simon Singh's fascinating book Fermat's Last Theorem. Serious passions have always been boiling around the insidious theorem, alluring with its apparent simplicity. The history of its 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, distinguished by great extravagance in life, who created the basis for Wiles' attack in 1955. Based on his ideas, Goro Shimura and Andre Weil a few years later (60-67) finally formulated the famous conjecture, having proved a significant part of which, Wiles obtained Fermat’s theorem as a corollary. The mysticism of the death story of the non-trivial Yutaka is associated with his stormy temperament: he hanged himself at the age of thirty-one due to unhappy love.

The entire long history of the mysterious theorem was accompanied by constant announcements about its proof, starting with Fermat himself. Constant errors in the endless stream of proofs befell not only amateur mathematicians, but also professional mathematicians. This led to the fact that the term "Fermatist", applied to those who proved Fermat's theorem, became a common noun. The constant intrigue with its proof sometimes led to funny incidents. So, when a gap was discovered in the first version of Wiles’ already widely publicized proof, a malicious inscription appeared at one of the New York subway stations: “I have found a truly wonderful proof of Fermat’s Last Theorem, but my train has arrived and I don’t have time to write it down.”

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

After completing graduate school, Wiles accepted a position at Princeton University, where he still works. He is married and has three daughters, two of whom were born "during the seven-year process of the first version of the proof." During these years, only Nada, Andrew's wife, knew that he was storming alone the most inaccessible and most famous peak of mathematics. It is to them, Nadya, Claire, Kate and Olivia, that Wiles’s famous final article “Modular elliptic curves and Fermat’s Last Theorem” in the central mathematical journal “Annals of Mathematics” is dedicated, where the most important mathematical works are published.

The events themselves around the proof unfolded quite dramatically. This exciting scenario could be called “fermatist – professional mathematician”.

Indeed, Andrew dreamed of proving Fermat's theorem since his youth. But, unlike the overwhelming majority of Fermatists, it was clear to him that for this it was necessary to master entire layers of the most complex mathematics. Moving towards his goal, Andrew graduates from the Faculty of Mathematics at the famous Cambridge University and begins to specialize in modern number theory, which is at the intersection with algebraic geometry.

The idea of storming 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, the Iwasawa theory, developed by Wiles himself and already familiar to him, which has deep historical roots, is chosen. This theory generalized Kummer's theory, historically the first serious mathematical theory to attack 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 a girl’s honor (pay attention, remembering the story with Taniyama, to the fatal role of beautiful ladies in the history of mathematics) .

Wiles is completely immersed in proof, even stopping participation in scientific conferences. And as a result of a seven-year retreat from the mathematical community at Princeton, in May 1993, Andrew put an end to his text - the job was done.

It was at this time that an excellent opportunity presented itself to notify the scientific world about his discovery - already in June a conference was to be held in his native Cambridge on precisely the desired topic. Three lectures at the Cambridge Institute by Isaac Newton excite not only the mathematical world, but also the general public. At the end of the third lecture, June 23, 1993, Wiles announces the proof of Fermat's Last Theorem. The proof is full of a whole bunch of new ideas, such as a new approach to the Taniyama-Shimura-Weil conjecture, a far advanced theory of Iwasawa, a new “deformation control theory” of Galois representations. The mathematical community is eagerly waiting for the text of the proof to be reviewed by experts in arithmetic algebraic geometry.

This is where the dramatic turn comes. Wiles himself, in the process of communicating with reviewers, discovers a gap in his evidence. The crack was caused by the “deformation control” mechanism he himself invented - the supporting structure of the proof.

The gap is revealed a couple of months later by Wiles' line-by-line explanation of his proof to Princeton faculty colleague Nick Katz. Nick Katz, having been on friendly terms with Andrew for a long time, recommends that he collaborate with the young promising English mathematician Richard Taylor.

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

And here's a new test. Not completed, but still very impressive, the result of Wiles’ work was reported by him to the International Congress of Mathematicians in Zurich at the end of August 1994. Wiles fights hard. Literally before the report, according to eyewitnesses, he was feverishly writing something else, trying to maximally improve the situation with the “sagging” evidence.

After this intriguing audience of the world's leading mathematicians, Wiles's report, the mathematical community “exhales joyfully” and sympathetically applauds: it’s okay, guy, no matter what happens, but he has advanced science, showing that in solving such an impregnable hypothesis one can successfully advance, which no one has ever done before I didn't even think about doing it. Another Fermatist, Andrew Wiles, could not take away the secret dream of many mathematicians about proving Fermat's theorem.

It is natural to imagine Wiles's condition at that time. Even the support and friendly attitude of his colleagues could not compensate for his state of psychological devastation.

And so, just a month later, when, as Wiles writes in the introduction to his final Annals article with the final proof, “I decided to take one last look at Eulerian 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 collaboration with Richard Taylor in the study of the Eulerian systems of Kolyvagin and Thain allowed the proof to be finalized in the form of two large papers in October.

Their publication, which filled the entire issue of the Annals of Mathematics, followed in November 1994. All this caused a new powerful information surge. The story of Wiles' proof received 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 assessment of his own work, Wiles noted that he had invented the mathematics of the future.

(I wonder if this is so? Let us just note that with all this information storm there was a sharp contrast with the almost zero information resonance in Russia, which continues to this day).

Let’s ask ourselves a question: what is the “internal kitchen” of obtaining outstanding results? After all, it is interesting to know how a scientist organizes his work, what he focuses on in it, and how he determines the priorities of his activities. What can be said about Andrew Wiles in this sense? And unexpectedly it turns out that in the modern era of active scientific communication and a collective style of work, Wiles had his own view on the style of working on super problems.

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

Such activity outside society, which did not involve direct scientific communication with colleagues even at conferences, seemed to contradict 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 closest peaks that could be conquered by modern mathematics in those years. At the same time, none of the specialists denied its enormous significance, and nominally it was in the “mainstream” of modern mathematics.

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

Wiles' 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, quite neutral. After the emotions and delight of the first time after the appearance of the landmark evidence subsided, everyone calmly continued their business. Specialists in arithmetic algebraic geometry slowly studied the “mighty proof” in their narrow circle, while the rest plowed their mathematical paths, diverging, as before, further and further from each other.

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

Then - again - you don’t want to get out of your cozy hole, 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 intrusion.

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

The subjective factors of the neutrality of the reaction of the mathematical world to the “event of the millennium” lie in quite prosaic reasons. The proof is indeed extraordinarily complex and lengthy. To a non-specialist in arithmetic algebraic geometry, it appears to consist of a layering of terminology and constructions of the most abstract mathematical disciplines. It seems that the author did not at all set a goal for him to be 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 synthetic nature of Wiles's proof, which used a variety of modern tools created by a large number of mathematicians in recent years.

But we must take into account that Wiles was not faced with the methodological task of explanation - he was constructing a new method. What worked in the method was precisely the synthesis of Wiles’s own brilliant ideas and a conglomerate of the latest results from various mathematical directions. And it was precisely such a powerful structure that rammed the impregnable problem. The proof was not an accident. The fact of its crystallization was fully consistent with both the logic of the development of science and the logic of knowledge. The task of explaining such a super-proof seems to be an absolutely independent, very difficult, although very promising problem.

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

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

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

For example, in classical number theory you will not find such long proofs as Wiles's. As number theorists say, “a proof should be a page long” (Wiles’s proof in collaboration with Taylor in the journal version takes 120 pages).

You also cannot exclude the factor of fear for the unprofessionalism of your assessment: by reacting, you take responsibility for assessing the evidence. How to do this when you don’t know this mathematics?

The position taken by direct specialists in number theory is characteristic: “... 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 the book by Paulo Ribenboim “Fermat’s Last Theorem for Amateurs” - the only one available today day to the source directly from Wiles' proof for the general reader.

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

At the Faculty of Mechanics and Mathematics of Moscow State University, however, proof enthusiasts appear. A remarkable mathematician and popular scientist Yu.P. Soloviev (untimely departed from us) initiates the translation of E. Knapp’s book on elliptic curves with the necessary material on the Taniyama-Shimura-Weil conjecture. Alexey Panchishkin, now working in France, gave lectures at the Faculty of Mechanics and Mathematics in 2001, which served as the basis for his corresponding part with Yu.I. Manin of the excellent book on modern number theory mentioned above (published in Russian translation by Sergei Gorchinsky with editing by Alexei Parshin in 2007).

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

There were also no public statements, much less discussions, on the part of famous Russian mathematicians regarding Wiles’ proof. There is a rather sharp discussion between the Russian V. Arnold (“a skeptic of the method of proof”) and the American S. Lang (“an enthusiast of the method of proof”), however, traces of it are lost in Western publications. In the Russian central mathematical press, during the time that has passed since the publication of Wiles' proof, there have been no publications on the topic of the proof. Perhaps the only publication on this topic was a translation of an article by Canadian mathematician Henry Darmon, even an incomplete version of the proof, in Advances in Mathematical Sciences in 1995 (it’s funny that the complete proof has already been published).

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

Nevertheless, the problem of proof adaptation, which extremely aggravates its applied potential, remained and remains very relevant. Today, the original highly specialized text of Wiles's article and the joint paper of Wiles and Taylor has already been adapted, although only for a fairly narrow circle of professional mathematicians. This was done in the mentioned book by Yu. Manin and A. Panchishkin. They managed to successfully smooth out a certain artificiality of the original proof. In addition, the American mathematician Serge Lang, an ardent promoter of Wiles' proof (who sadly 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 particularly striking features that creates 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 ordinary sense. It does not contain natural geometric images to which one could attach for a better understanding of the text. Super-powerful “terminologized” abstract algebra and “advanced” number theory purely psychologically undermine the ability to perceive proof even for a qualified mathematical reader.

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

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

The famous Russian mathematician of the middle of the last century, former dean of the Faculty of Mechanics and Mathematics, V.V., spoke very clearly about the powerful, but not yet applied, mathematics. Golubev:

“... according to the witty remark of F. Klein, many departments of mathematics are similar to those exhibitions of the latest models of weapons that exist at companies that manufacture weapons; with all the wit put in by the inventors, it often happens that when a real war begins, these new products turn out to be unusable for one reason or another... The modern teaching of mathematics presents exactly the same picture; students are given very advanced and powerful means of mathematical research into their hands..., but then 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 it is the creative will of man. At one time A.P. Chekhov said that if in the first act of a play there is a gun hanging on the stage, then it is necessary that at least in the third act it be fired. This remark 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’ proof represents 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 branches 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 the question: is it possible to describe Wiles’ proof in sufficiently accessible terms for a wide interested audience?

From the point of view of experts, this is an absolute utopia. But let’s try anyway, guided by the simple consideration that Fermat’s theorem is a statement only about integer points of our ordinary three-dimensional Euclidean space.

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

Wiles finds the optimal mechanism for recalculating integer points and testing them to satisfy the equation of Fermat’s theorem (after introducing the necessary definitions, such a recalculation will precisely correspond to the so-called “modularity property of elliptic curves over the field of rational numbers”, described by the Taniyama-Shimura-Weil conjecture).

The recalculation mechanism is optimized with the help of a remarkable discovery by the German mathematician Gerhard Frey, who connected a potential solution of the Fermat equation with an arbitrary exponent with another, completely different equation. This new equation is given by a special curve (called Frey's 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 associated with every solution of Fermat’s equation, that is, numbers satisfying the relation

the above curve. Now it remains to show that such curves do not exist for .

Frey’s invention at the time of Wiles’s “start” was quite fresh (the year 1985) and also echoed the relatively recent approach of the French mathematician Helleguarche (the 1970s), 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, we can say that Wiles invents tools (special algebraic constructions) to control this recalculation. As a matter of fact, this subtle toolkit of Wiles constitutes the central core and main complexity of the proof. It is in the manufacture of these instruments that Wiles's main sophisticated algebraic discoveries, which are so difficult to comprehend, arise.

But still, the most unexpected effect of the proof, perhaps, is the sufficiency of using only one “Freevian” curve, represented by a completely simple, almost “school” dependence.

Surprisingly, using only one such curve is sufficient to test all points in three-dimensional Euclidean space with integer coordinates to see if they satisfy Fermat's Last Theorem with an arbitrary exponent.

In other words, using just one curve (though it has a specific form), understandable to an ordinary high school student, turns out to be equivalent to constructing an algorithm (program) for sequential recalculation of whole points of ordinary three-dimensional space. And not just a recalculation, but a recalculation with simultaneous testing of the whole point for “its satisfaction” with Fermat’s equation.

It is here that the phantom of Pierre de Fermat himself arises, since with such a recalculation what is usually called Fermat’s “Ferma’t descent,” or 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 he could well “see” the geometric idea of its proof.

The most important thing here is that these tools are “minimal”, i.e. they cannot be simplified. Although this “minimalism” in itself is very difficult. And it was Wiles’s awareness of this non-trivial “minimality” that became the decisive final step of the proof. This was exactly the “outbreak” on September 19, 1994.

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

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

Now let’s move directly to the virtual experiment and try to “dig” into the thoughts of mathematician-lawyer Pierre de Fermat.

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

You 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 transforms 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 along a plane and “winding” entire points on this plane around itself. It is important that this rolling should not be arbitrary, but “periodic” (mathematicians also say “cyclotomic”). The periodicity of rolling means that the linear and angular velocity vectors of a sphere rolling in the most general manner after a certain fixed time (period) are repeated in magnitude and direction. This periodicity is similar to the periodicity of the linear speed of rolling a circle along a straight line, modeling the “small” Fermat equation.

Accordingly, the “large” Fermat equation takes on the meaning of the law of conservation of the above-mentioned 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 all the 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 - there is no need to find such a diagonal, since time is one-dimensional and there is no reason to take a diagonal. Therefore, the degree of a variable in the equation of Fermat’s little theorem can be arbitrary.

So, quite unexpectedly, we get a bridge to the “physicalization” of Fermat’s great theorem, that is, to the appearance of its physical meaning. How can one not remember that Fermat was no stranger to physics.

By the way, the experience of physics also shows that the laws of conservation 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 of Newtonian mechanics, known from school.

From the point of view of the above “physical” interpretation of Fermat’s last theorem, the property of “minimality” corresponds to the minimality of the 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 (minimal in complexity) recalculation, both geometrically and algebraically, is represented by the rolling of a sphere on a plane, since a sphere and a plane are “minimal,” as we completely understand, two-dimensional geometric objects.

The whole complexity, which at first glance is missing, lies in the fact that an accurate description of such a seemingly “simple” movement of the sphere is not at all easy. The fact 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 caused by non-trivial combinations (compositions) of the linear and angular motion of the sphere - see Fig. 1.

It is 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’ algebraic constructions are required.

In the geometric interpretation shown in Fig. 1, the linear movement of the center of the sphere “counts” whole 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 cannot be immediately “seen” in the arbitrary rolling of the sphere along the plane. It is the rotational motion that corresponds to the hidden symmetries of Euclidean space mentioned above.

The Frey curve introduced above precisely “encodes” the most aesthetically beautiful recalculation of whole points in space, reminiscent of movement along a spiral staircase. Indeed, if you follow the curve that a certain point on the sphere sweeps in one period, you will find that our marked point sweeps 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 plot of the "minimum" of the (i.e.) Frey curve. This is the schedule 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 for DNA-encoding constructs from Wiles' proof is used in Singh's book Fermat's Last Theorem.

Let us emphasize once again that the decisive point in our interpretation is the fact that the analogue of the conservation law for Fermat’s little theorem (its degree can be arbitrarily large) turns out to be the equation of Fermat’s Great Theorem precisely in the case .

It is this effect of “minimality of the degree of conservation law for the rolling of a sphere on a plane” that corresponds to the statement of Fermat’s Last Theorem.

Now let's build a bridge to modern physics. The geometric image of Wiles's proof proposed here is very close to the geometry of modern physics, which is trying to get to the mystery of the nature of gravity - the quantum general theory of relativity. To confirm this, at first glance unexpected, interaction between Fermat’s Last Theorem and Big Physics, let’s imagine that the rolling sphere is massive and “pushes” the plane beneath it. The interpretation of this “pushing” in Fig. 3 is strikingly reminiscent of 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 actually observe “quantum gravity” with our own eyes!

It is on this major “unifying” physico-mathematical note that we will end 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 in one way or another use the constructions and logic of Wiles’ proof. It is simply impossible to bypass all this due to the mentioned “minimality property” of Wiles’ mathematical tools used for the proof. In our “geometric-dynamical” interpretation of this proof, this “minimality property” provides the “minimum necessary conditions” for a correct (i.e., “convergent”) construction of a testing algorithm.

On the one hand, this is a huge disappointment for amateur farmers (if, of course, they find out about it; as they say, “the less you know, the better you sleep”). On the other hand, the natural “unsimplification” of Wiles’s proof formally makes life easier for professional mathematicians - they may not read periodically emerging “elementary” proofs from amateur mathematics, citing the lack of correspondence with Wiles’s proof.

The general conclusion is that both need to “strain” and understand this “savage” proof, essentially comprehending “all mathematics.”

What else is important not to miss when summing up this entire unique story that we have witnessed? The strength of Wiles's proof is that it is not simply a formal logical argument, but represents 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 fundamentally important that when we look down from the height of the skyscraper at Wiles’s proof, we will see all the previous mathematics. The pathos is 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 evidence. All that remains is “just nothing” - just understand it and learn to apply it.

I wonder what our contemporary hero Wiles is doing today? There is no special news about Andrew. He, naturally, received various awards and prizes, including the famous German Wolfskehl Prize, which was depreciated during the first civil war. In 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” conjecture - recently formulated (by Masser and Oesterle in 1986) and considered the most important problem in number theory today (it is the “problem of the century” in the words of Serge Lang ).

Much more information about Wiles' co-author on the final part of the proof, Richard Taylor. He was one of the four authors of the proof of the full Taniyama-Shmura-Weil conjecture and was a strong contender for the Fields Medal at the 2002 Chinese Mathematical Congress. However, he did not receive it (then 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 a considerable number of remarkable works during this time. And 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 dream up a little. Perhaps the time will come when mathematics courses in universities, and even in schools, will be adjusted to Wiles' methods of proof. This means that Fermat's Last Theorem will become not only a model mathematical problem, but also a methodological model for teaching mathematics. Using her example, it will be possible to study, in fact, all the main branches of mathematics. Moreover, future physics, and maybe even biology and economics, will begin to rely 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 Lang included the main constructions of Wiles' proof in the third edition of his classic manual on algebra. The Russians Yuri Manin and Alexey Panchishkin go even further in the aforementioned new edition of their “Modern Theory of Numbers,” setting out in detail the proof itself in the context of modern mathematics.

And how can one not exclaim now: Fermat’s great theorem is “dead” - long live Wiles’ method!

A sensational message was broadcast that the Omsk scientist Alexander Ilyin had found a simple proof of Fermat's Last Theorem. The news about this even made it onto television. However, a professional analysis of the evidence revealed a gross error in it.

The theorem was formulated by the famous 17th century mathematician Pierre Fermat. It is that the equation

x n + y n = z n

Has no integer solutions for n> 2. In the margins of the book, Fermat left a note that he allegedly found a surprisingly elegant proof of this theorem. However, for more than three centuries no one was able to find this proof. Only in 1994, the Great Theorem was proved by the English mathematician Andrew Wiles, and the proof took more than a hundred pages of mathematical calculations.

Wiles' proof uses mathematical apparatus developed only in the 20th century. Therefore, many mathematics lovers continue to search for the legendary simple proof using elementary school mathematics. With enviable regularity, such evidence is received by a variety of scientific organizations. Sometimes the authors of these opuses are unfamiliar even with the basics of mathematical culture and mix mathematical calculations with lengthy philosophical reasoning. Experts jokingly call such would-be mathematicians “farmatists.” There is even a poem dedicated to attempts to prove Fermat's Last Theorem.

How does the current case differ from all previous ones? Because this time the elementary proof of Fermat’s theorem was published by a prominent scientist, Academician Ilyin, former chief designer of the Polet aerospace association. According to media reports, his proof was checked by several familiar scientists, in particular, academician Leonid Gorynin and professor Sergei Chukanov *), and concluded that they did not find any flaws in Ilyin’s argumentation. And although neither the author nor the reviewers are experts in number theory, the status allowed Academician Ilyin to convene press conferences in Omsk and Moscow, where he presented his proof to journalists.

On August 22, sensational evidence was published in Novaya Gazeta. It was also reported on television. Some media outlets (including Novaya Gazeta) reported the evidence as an immutable fact. Others, such as the analytical agency Glavred, spoke with some caution. However, only Radio Liberty turned to mathematicians from the Moscow Center for Continuing Mathematical Education with a request to study the published solution of Fermat’s theorem. Here is a quote from the response received:

“Any tenth grader with a grade above C in mathematics will immediately reproduce for you the formula for the ratio of the sides of a triangle z 2 = x 2 + y 2 — 2xy cos( b). Let's look at the expression. At 60° b b) is not an integer. Which means z is inevitably so for integer values x And y».

However, from the fact that cos( b) is not an integer, it does not at all follow that the product 2 is such xy cos( b). Let's say, when b= arccos(1/4) (which is approximately equal to 75 degrees, i.e. falls within the required range from 60 to 90 degrees) cos( b) = 1/4, and if at least one of the numbers x And y even, then 2 xy cos( b) will be integer.

Once discovered, this error becomes quite obvious at the level of a school mathematics course. According to professional mathematicians, this case can serve as a clear illustration of the fact that sensational discoveries published in circumvention of the mandatory peer review system adopted in science most often turn out to be misunderstandings.

*) On the morning of August 26, the editors received a letter from Prof. Sergei Nikolaevich Chukanov with a request to publish it on the website. The editors readily comply with this request.

Dear editors of the Elements project!

I consider it necessary to comment on the message of Alexander Sergeev “The sensation around Fermat’s theorem turned out to be a misunderstanding” dated August 25, 2005 on your website: “As the media report, his proof was checked by several familiar scientists: Professor Sergei Chukanov, and they gave a conclusion about what was not found in the argument Ilyin's flaws." This misunderstanding is aggravated by the fact that I first became acquainted with the “evidence” from an article by Anna Melekhova on the website of the National News Agency.

In the article, the “proof” is based on the proposition: “since cos a on the interval (11) takes only irrational values,” which indicates that the author of this “proof” lacks elementary mathematical knowledge. I have not found any proofs of Fermat’s Last Theorem by Alexander Ilyin published in peer-reviewed publications.

Sincerely,
Sergey Nikolaevich Chukanov

We regret that the reputation of Prof. Chukanova could have suffered due to incorrect media publications, and we share his bewilderment.

This is exactly the amount that British mathematician Andrew Wiles will receive, who in 1994 presented a proof of Fermat's Last Theorem. The decision of the International Mathematical Union and the European Mathematical Society to award him the Abel Prize, sometimes called the Nobel Prize for mathematicians, was announced by the President of the Norwegian Academy of Sciences and Letters, Ole Sejersted. reportedon the official website of the award.

“The new ideas introduced by Wiles opened the door for subsequent breakthroughs,” said Jon Rognes, head of the Abel Committee. “Few mathematical problems have such a rich scientific history and such spectacular proof as Fermat’s Last Theorem.”

The great French mathematician Pierre Fermat always made his notes in the margins of the mathematical treatises he read and formulated problems and theorems that came to mind there. He wrote down his Great Theorem, which is sometimes called the Last Theorem, with a note that the ingenious proof he found of this theorem was too long to be placed in the margins of the book:

“On the contrary, it is impossible to decompose a cube into two cubes, a biquadrate into two biquadrates, and in general no power greater than a square into two powers with the same exponent. I found a truly wonderful proof of this, but the margins of the book are too narrow for it.”

Original text (Latin)

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

This was in 1637, when musketeers were galloping around France with might and main, stealing diamond pendants and killing each other in duels.

What this theorem is and how it is formulated, how difficult it is to prove it, is best told in a fascinating film by the wonderful Soviet documentary director Semyon Wartburg.

Attempts to prove WTF began almost immediately after its “discovery.” Euler, Dirichlet, Legendre and other professional mathematicians and amateurs struggled with this problem. Ernst Kummer also created modern number theory.

David Hilbert, in his report “Mathematical Problems” at the II International Congress of Mathematicians (1900), spoke about the WTF as follows:

“The problem of proving this undecidability is a striking example of the stimulating influence that a special and at first glance insignificant problem can have on science. For, prompted by Fermat's problem, Kummer came to the introduction of ideal numbers and to the discovery of the theorem on the unique decomposition of numbers in cyclotomic fields into ideal prime factors - a theorem which, thanks to the generalizations to any algebraic number domain obtained by Dedekind and Kronecker, is now central to modern theory of numbers and whose significance goes far beyond the theory of numbers into the field of algebra and function theory.”

It is the creation of number theory that can be called Fermat’s main posthumous contribution to the development of mathematics. “God created the integers, and everything else is the work of man,” is how Leopold Kronecker described the role of this theory. Pythagoras believed that the entire Universe consists of integers. In any case, man is exploring the secrets of the Universe so far using purely digital, discrete methods.

Andrew Wiles, Knight Commander of the Order of the British Empire - English and American mathematician, head of the department of mathematics at Princeton University, member of the scientific council of the Clay Institute of Mathematics, learned about WTF when he was only ten years old. Armed with a school textbook, he tried to surpass Euler and Dirichlet, but, of course, he did not succeed.

He calmed down, although as subsequent events showed, not for long, he received a higher education and took up other problems. Specifically, the so-called elliptic curves. He soon became acquainted with the work in this area of the Japanese Shimura and Takayama, as well as the Frenchman Weil. And then with the proof of the American mathematician Ken Ribet that these works are directly related to the WTF.

Then the “old love” rose from the depths of Wiles’ subconscious and for seven years he tried to finish off “Fermat`s last theorem” - this is the name of WTF in his work, for which he, in fact, received his honestly earned 700 thousand dollars.

Then all kinds of awards began to pour in on him, including the conferment of a high knighthood.

Since Wiles' proof contains 130 pages of extremely complex mathematical text, it took the Norwegian Academy of Sciences quite a few years to verify its correctness.

But even now with the WTF and its proof, not everything is so simple and clear.

The army of thousands of “fermatists”, that is, fans of the endless, but extremely absorbing proof of her fidelity or infidelity, is outraged and demands the continuation of the show.

Meanwhile, the passion for proving WTF is extremely dangerous for “romantic young men.” Here is what one of the indignant “farmatists” writes on an Internet forum:

“The mathematical community seems to have unconditionally accepted the fact that the long-awaited proof of the WTF has indeed been found. However, as a member of the mathematical community, I am in the unfortunate position of discussing or attesting to the validity of this proof. I have been told that the amount of time required to gain knowledge sufficient to criticize evidence is measured in years. After a brief review of the mathematical apparatus used, I was convinced that this was correct. Therefore, an awkward situation was created when the solution to such a classic problem appeared in a form understandable only to the most sophisticated experts. It is for this reason that I am writing this work. I hope that this work can serve as an initial reference for anyone interested in studying the necessary information for their own verification of the correctness of the WTF proof.”

I deliberately preserve the style and grammatical errors and typos of the original to show the typical level of such fans.

More experienced mathematical enthusiasts are also trying to understand Wiles’s proof:

“In my opinion, this is the case when understanding the proof is more difficult than detecting the error. Therefore, I would be careful not to call Wiles “the great Wiles” for now. - writes one of them.

“By the way, I also read up to that point about what “conductors” are and what “modular forms” are, but I didn’t understand anything at all about the Kolyvagin-Flach method. And in Wiles’s proof this is just the beginning, the starting point!” - the other one answers him.

Professionals are also digging into Wiles’ 130-page work:

“After looking at this article, I see again: the Ribet-Frey-Wiles proof relies on the fact that when decomposing the discriminant of the Frey curve ... into prime numbers, the power of two is not divisible by the exponent of the Fermat equation, and the remaining powers are divisible. If the denominator did not have 256, then the proof would not be suitable. It would be nice if someone could clarify this: why is there 256, and is it possible to generalize Fermat’s Theorem to Ribet’s proof? Or does 256 have nothing to do with it at all?”

A number of skeptics simply deny Wiles any significant contribution to the epochal proof:

“Taniyama saw (noted), formulated a hypothesis, Wiles proved the correctness of the hypothesis. Also, earlier there seemed to be a proof that the validity of Taniyama’s hypothesis implies the validity of Fermat’s theorem. From where I conclude that the resulting proof is a pure game of formulas and accidents.”

But other experts are closer to the truth, who see in the long-awaited proof of WTF the possibility of further promising discoveries:

“But this, I suppose, is not so. This is not accidental, and if there are no errors, then this theory reflects some algebraic properties of trinomials. Fermat's theorem is formulated for triplets of numbers, and the elliptic functions by which it is proved appear when solving a differential equation. Behind this may lie the remarkable properties of numbers and the unity of algebra. I think if anyone could clarify these connections, it would be very cool.”

And finally, the conclusion to which I, as a mathematical engineer, am ready to subscribe:

“I can’t agree with your words. In order to “stupidly see” this property, one had to work hard. But you are right in the sense that it seems that everything was done here without understanding the true reasons. I think that the one who understands why all this is arranged this way and why elliptic functions have such properties, and the one who can explain this, will make a greater contribution to mathematics than, for example, the same Wiles. Therefore, looking for a parallel proof of Fermat’s Theorem is extremely useful: if this is found, then the reasons for the properties of elliptic functions may be revealed, and if new methods are also invented, then this will be a breakthrough in mathematics, perhaps the best in the last 50 years. But of course, new methods are needed, without them it will be just an explanation, not a new discovery.”

So, Fermat's Last Theorem is finally proven. But there is no reason for die-hard fermatists to lay down their mathematical weapons. The scientific community is hungry for a simpler and more general proof.

Perhaps a corresponding bonus will be invented for this task. After all, to paraphrase Leopold Kronecker, God only came up with integers, and it is our task to correctly “sort” them. +

Mathematician Andrew Wiles received the Abel Prize for his proof of Fermat's theorem

An honorary award, called the "Nobel Prize for mathematicians", was awarded to him for his proof of Fermat's Last Theorem in 1994

OSLO, March 15. /Corr. TASS Yuri Mikhailenko/. Briton Andrew Wiles has been announced as the winner of the Abel Prize, awarded by the Norwegian Academy of Sciences. The honorary award, often called the “Nobel Prize for mathematicians,” was awarded to him for his proof of Fermat’s Last Theorem in 1994, which “launched a new era in number theory.”
“The new ideas introduced by Wiles opened up the possibility of further breakthroughs,” said Abel Committee Chairman Jon Rognes. “Few mathematical problems have as rich a scientific history and as spectacular a proof as Fermat’s Last Theorem.”
Sir Andrew's scientific journey
In comments to the Norwegian Telegraph Bureau, Rognes also clarified that the proof of the famous theorem was just one of the reasons why Wiles was chosen among the candidates nominated for the prize this year.
“To solve a theorem that could not be proven for 350 years, he used the approaches of two modern branches of mathematical science, studying, in particular, semi-stable elliptic curves,” Rognes told reporters. “Such mathematics is used, for example, in elliptic cryptography, with the help of which security data on payments made using plastic cards."
The scientist, who turns 63 next month, was educated at Oxford and Cambridge universities. His father was an Anglican clergyman and was professor of theology at Cambridge for more than 20 years. Wiles himself worked in the United States for 30 years, teaching at Princeton University, and from 2005 to 2009 headed the mathematics department there. He currently works in Oxford. He has won a dozen mathematical prizes, and for his scientific achievements he was also knighted by Queen Elizabeth II of Great Britain.
Deceptive simplicity
The peculiarity of the theorem, formulated by the Frenchman Pierre Fermat (1601 - 1665), is in a deceptively simple formulation: the equation “A to the power of n plus B to the power of n is equal to C to the power of n” has no natural solutions if the number n is greater than two. At first glance, it suggests a fairly simple proof, but in reality this turns out to be completely different.
Wiles himself admitted in numerous interviews that the theorem intrigued him at the age of 10. Even then, it was easy for him to understand the conditions of the problem, and he was haunted by the fact that for three centuries not a single mathematician had been able to solve it. The childhood hobby has not faded over the years. Having already made a scientific career, Wiles spent many years struggling with the solution in his free time, but did not advertise it, since among his colleagues, a passion for Fermat’s theorem was considered bad manners. He proposed his proof, based on the hypothesis of two Japanese scientists, and published it in 1993, but a few months later an error was discovered in his calculations.
For more than a year, Wiles, together with his students, tried to correct it, almost giving up in the end, but ultimately still found a proof that was recognized as correct. At the same time, the supposedly existing simple and elegant proof, which Fermat himself mentioned, has not yet been found.
Who was Henrik Abel
In 2014 and 2009, the Abel Prize laureates were students of the Russian mathematical school - Yakov Sinai and Mikhail Gromov, respectively. The award is named after the famous Norwegian Niels Henrik Abel. He became the founder of the theory of elliptic functions and made significant contributions to the theory of series.
In honor of the 200th anniversary of the birth of the scientist, who lived only 26 years, the Norwegian government in 2002 allocated 200 million kroner (about $23.4 million at current exchange rates) to establish the Abel Foundation and the Abel Prize. It is intended not only to celebrate the merits of outstanding mathematicians, but also to contribute to the growing popularity of this scientific discipline among young people.
Today, the cash component of the award is 6 million crowns ($700 thousand). The official award ceremony is scheduled to take place on May 24. The honorary award will be presented to the laureate by the heir to the Norwegian throne, Prince Haakon Magnus.