It is unlikely that at least one year in the life of our editorial office passed without it receiving a good dozen proofs of Fermat's theorem. Now, after the “victory” over it, the flow has subsided, but has not dried up.

Of course, not to dry it completely, we publish this article. And not in his own defense - that, they say, that's why we kept silent, we ourselves have not matured yet to discuss such complex problems.

But if the article really seems complicated, look at the end of it right away. You will have to feel that the passions have calmed down temporarily, the science is not over, and soon new proofs of new theorems will be sent to the editors.

It seems that the 20th century was not in vain. First, people created a second Sun for a moment by detonating a hydrogen bomb. Then they walked on the moon and finally proved the notorious Fermat's theorem. Of these three miracles, the first two are on everyone's lips, for they have had enormous social consequences. On the contrary, the third miracle looks like another scientific toy - on a par with the theory of relativity, quantum mechanics and Gödel's theorem on the incompleteness of arithmetic. However, relativity and quanta led physicists to hydrogen bomb, and the research of mathematicians filled our world with computers. Will this string of miracles continue into the 21st century? Is it possible to trace the connection between the next scientific toys and revolutions in our everyday life? Does this connection allow us to make successful predictions? Let's try to understand this using the example of Fermat's theorem.

Let's note for a start that she was born much later than her natural term. After all, the first special case of Fermat's theorem is the Pythagorean equation X 2 + Y 2 = Z 2 , relating the lengths of the sides of a right triangle. Having proved this formula twenty-five centuries ago, Pythagoras immediately asked himself the question: are there many triangles in nature in which both legs and hypotenuse have an integer length? It seems that the Egyptians knew only one such triangle - with sides (3, 4, 5). But it is not difficult to find other options: for example (5, 12, 13) , (7, 24, 25) or (8, 15, 17) . In all these cases, the length of the hypotenuse has the form (A 2 + B 2), where A and B are coprime numbers of different parity. In this case, the lengths of the legs are equal to (A 2 - B 2) and 2AB.

Noticing these relationships, Pythagoras easily proved that any triple of numbers (X \u003d A 2 - B 2, Y \u003d 2AB, Z \u003d A 2 + B 2) is a solution to the equation X 2 + Y 2 \u003d Z 2 and sets a rectangle with mutually simple side lengths. It is also seen that the number of different triples of this sort is infinite. But do all solutions of the Pythagorean equation have this form? Pythagoras was unable to prove or disprove such a hypothesis and left this problem to posterity without drawing attention to it. Who wants to highlight their failures? It seems that after this the problem of integral right-angled triangles lay in oblivion for seven centuries - until a new mathematical genius named Diophantus appeared in Alexandria.

We know little about him, but it is clear that he was nothing like Pythagoras. He felt like a king in geometry and even beyond - whether in music, astronomy or politics. The first arithmetic connection between the lengths of the sides of a harmonious harp, the first model of the Universe from concentric spheres carrying planets and stars, with the Earth in the center, and finally, the first republic of scientists in the Italian city of Crotone - these are the personal achievements of Pythagoras. What could Diophantus oppose to such successes - a modest researcher of the great Museum, which has long ceased to be the pride of the city crowd?

Only one thing: a better understanding ancient world numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet own the positional notation system big numbers but he knew what negative numbers and probably spent many hours thinking about why the product of two negative numbers is positive. The world of integers was first revealed to Diophantus as a special universe, different from the world of stars, segments or polyhedra. The main occupation of scientists in this world is solving equations, a true master finds all possible solutions and proves that there are no other solutions. This is what Diophantus did quadratic equation Pythagoras, and then he thought: does at least one solution have a similar cubic equation X 3 + Y 3 = Z 3?

Diophantus failed to find such a solution; his attempt to prove that there are no solutions was also unsuccessful. Therefore, drawing up the results of his work in the book "Arithmetic" (it was the world's first textbook on number theory), Diophantus analyzed the Pythagorean equation in detail, but did not hint at a word about possible generalizations of this equation. But he could: after all, it was Diophantus who first proposed the notation for the powers of integers! But alas: the concept of “task book” was alien to Hellenic science and pedagogy, and publishing lists of unsolved problems was considered an indecent occupation (only Socrates acted differently). If you can't solve the problem - shut up! Diophantus fell silent, and this silence dragged on for fourteen centuries - until the onset of the New Age, when interest in the process of human thinking was revived.

Who didn’t fantasize about anything at the turn of the 16th-17th centuries! The indefatigable calculator Kepler tried to guess the connection between the distances from the Sun to the planets. Pythagoras failed. Kepler's success came after he learned how to integrate polynomials and other simple functions. On the contrary, the dreamer Descartes did not like long calculations, but it was he who first presented all points of the plane or space as sets of numbers. This audacious model reduces any geometric problem about figures to some algebraic problem about equations - and vice versa. For example, integer solutions of the Pythagorean equation correspond to integer points on the surface of a cone. The surface corresponding to the cubic equation X 3 + Y 3 = Z 3 looks more complicated, its geometric properties did not suggest anything to Pierre Fermat, and he had to pave new paths through the wilds of integers.

In 1636, a book by Diophantus, just translated into Latin from a Greek original, fell into the hands of a young lawyer from Toulouse, accidentally surviving in some Byzantine archive and brought to Italy by one of the Roman fugitives at the time of the Turkish ruin. Reading an elegant discussion of the Pythagorean equation, Fermat thought: is it possible to find such a solution, which consists of three square numbers? There are no small numbers of this kind: it is easy to verify this by enumeration. What about big decisions? Without a computer, Fermat could not carry out a numerical experiment. But he noticed that for each "large" solution of the equation X 4 + Y 4 = Z 4, one can construct a smaller solution. So the sum of the fourth powers of two integers is never equal to the same power of the third number! What about the sum of two cubes?

Inspired by the success for degree 4, Fermat tried to modify the "method of descent" for degree 3 - and succeeded. It turned out that it was impossible to compose two small cubes from those single cubes into which a large cube with an integer length of an edge fell apart. The triumphant Fermat made a brief note in the margins of Diophantus's book and sent a letter to Paris with a detailed report of his discovery. But he did not receive an answer - although usually mathematicians from the capital reacted quickly to the next success of their lone colleague-rival in Toulouse. What's the matter here?

Very simple: to mid-seventeenth century, arithmetic went out of fashion. The great successes of the Italian algebraists of the 16th century (when polynomial equations of degrees 3 and 4 were solved) did not become the beginning of a general scientific revolution, because they did not allow solving new bright problems in adjacent fields of science. Now, if Kepler could guess the orbits of the planets using pure arithmetic ... But alas, this required mathematical analysis. This means that it must be developed - up to complete triumph mathematical methods in natural science! But analysis grows out of geometry, while arithmetic remains a field of play for idle lawyers and other lovers of the eternal science of numbers and figures.

So, Fermat's arithmetic successes turned out to be untimely and remained unappreciated. He was not upset by this: for the fame of a mathematician, the facts of differential calculus, analytic geometry and probability theory were revealed to him for the first time. All these discoveries of Fermat immediately entered the golden fund of the new European science, while number theory faded into the background for another hundred years - until it was revived by Euler.

This "king of mathematicians" of the 18th century was a champion in all applications of analysis, but he did not neglect arithmetic either, since new methods of analysis led to unexpected facts about numbers. Who would have thought that the infinite sum of inverse squares (1 + 1/4 + 1/9 + 1/16+…) is equal to π 2 /6? Who among the Hellenes could have foreseen that similar series would make it possible to prove the irrationality of the number π?

Such successes forced Euler to carefully reread the surviving manuscripts of Fermat (fortunately, the son of the great Frenchman managed to publish them). True, the proof of the “big theorem” for degree 3 has not been preserved, but Euler easily restored it just by pointing to the “descent method”, and immediately tried to transfer this method to the next prime degree - 5.

It wasn't there! In Euler's reasoning appeared complex numbers, which Fermat managed not to notice (such is the usual lot of discoverers). But the factorization of complex integers is a delicate matter. Even Euler did not fully understand it and put the "Fermat problem" aside, in a hurry to complete his main work - the textbook "Fundamentals of Analysis", which was supposed to help every talented young man to stand on a par with Leibniz and Euler. The publication of the textbook was completed in St. Petersburg in 1770. But Euler did not return to Fermat's theorem, being sure that everything that his hands and mind touched would not be forgotten by the new scientific youth.

And so it happened: the Frenchman Adrien Legendre became Euler's successor in number theory. At the end of the 18th century, he completed the proof of Fermat's theorem for degree 5 - and although he failed for large prime powers, he compiled another textbook on number theory. May its young readers surpass the author in the same way that the readers of the Mathematical Principles of Natural Philosophy surpassed the great Newton! Legendre was no match for Newton or Euler, but there were two geniuses among his readers: Carl Gauss and Evariste Galois.

Such a high concentration of geniuses was facilitated by the French Revolution, which proclaimed the state cult of Reason. After that, every talented scientist felt like Columbus or Alexander the Great, able to discover or conquer a new world. Many succeeded, which is why in the 19th century scientific and technological progress became the main driver of the evolution of mankind, and all reasonable rulers (starting with Napoleon) were aware of this.

Gauss was close in character to Columbus. But he (like Newton) did not know how to captivate the imagination of rulers or students with beautiful speeches, and therefore limited his ambitions to the sphere of scientific concepts. Here he could do whatever he wanted. For example, the ancient problem of the trisection of an angle for some reason cannot be solved with a compass and straightedge. With the help of complex numbers depicting points of the plane, Gauss translates this problem into the language of algebra - and obtains a general theory of the feasibility of certain geometric constructions. Thus, at the same time, a rigorous proof of the impossibility of constructing a regular 7- or 9-gon with a compass and a ruler appeared, and such a way of constructing a regular 17-gon, which the wisest geometers of Hellas did not dream of.

Of course, such success is not given in vain: one has to invent new concepts that reflect the essence of the matter. Newton introduced three such concepts: flux (derivative), fluent (integral) and power series. They were enough to create mathematical analysis and the first scientific model of the physical world, including mechanics and astronomy. Gauss also introduced three new concepts: vector space, field, and ring. A new algebra grew out of them, subordinating Greek arithmetic and the theory of numerical functions created by Newton. It remained to subordinate the logic created by Aristotle to algebra: then it would be possible to prove the deducibility or non-derivability of any scientific statements from this set of axioms with the help of calculations! For example, does Fermat's theorem derive from the axioms of arithmetic, or does Euclid's postulate of parallel lines derive from other axioms of planimetry?

Gauss did not have time to realize this daring dream - although he advanced far and guessed the possibility of the existence of exotic (non-commutative) algebras. Only the daring Russian Nikolai Lobachevsky managed to build the first non-Euclidean geometry, and the first non-commutative algebra (Group Theory) was managed by the Frenchman Evariste Galois. And only much later than the death of Gauss - in 1872 - the young German Felix Klein guessed that the variety of possible geometries can be brought into one-to-one correspondence with the variety of possible algebras. Simply put, every geometry is defined by its symmetry group - while general algebra studies all possible groups and their properties.

But such an understanding of geometry and algebra came much later, and the assault on Fermat's theorem resumed during Gauss's lifetime. He himself neglected Fermat's theorem out of the principle: it is not the king's business to solve individual problems that do not fit into a bright scientific theory! But the students of Gauss, armed with his new algebra and the classical analysis of Newton and Euler, reasoned differently. First, Peter Dirichlet proved Fermat's theorem for degree 7 using the ring of complex integers generated by the roots of this degree of unity. Then Ernst Kummer extended the Dirichlet method to EVERYTHING simple degrees(!) - so it seemed to him rashly, and he triumphed. But soon a sobering up came: the proof passes flawlessly only if every element of the ring is uniquely decomposed into prime factors! For ordinary integers, this fact was already known to Euclid, but only Gauss gave its rigorous proof. But what about the whole complex numbers?

According to the “principle of the greatest mischief”, there can and SHOULD occur an ambiguous factorization! As soon as Kummer learned to calculate the degree of ambiguity by methods of mathematical analysis, he discovered this dirty trick in the ring for the degree of 23. Gauss did not have time to learn about this version of exotic commutative algebra, but Gauss's students grew up in place of another dirty trick a new beautiful Theory of Ideals. True, this did not help much in solving Fermat's problem: only its natural complexity became clearer.

Throughout the 19th century, this ancient idol demanded more and more sacrifices from its admirers in the form of new complex theories. It is not surprising that by the beginning of the 20th century, believers became discouraged and rebelled, rejecting their former idol. The word "fermatist" has become a swear word among professional mathematicians. And although a considerable prize was assigned for the complete proof of Fermat's theorem, but its applicants were mostly self-confident ignoramuses. The strongest mathematicians of that time - Poincaré and Hilbert - defiantly eschewed this topic.

In 1900, Hilbert did not include Fermat's Theorem in the list of twenty-three major problems facing the mathematics of the twentieth century. True, he included in their series the general problem of the solvability of Diophantine equations. The message was clear: follow the example of Gauss and Galois, create general theories new mathematical objects! Then one fine (but not predictable in advance) day, the old splinter will fall out by itself.

This is how the great romantic Henri Poincaré acted. Neglecting many "eternal" problems, all his life he studied the SYMMETRIES of various objects of mathematics or physics: either functions of a complex variable, or trajectories of motion of celestial bodies, or algebraic curves or smooth manifolds (these are multidimensional generalizations of curved lines). The motive for his actions was simple: if two different objects have similar symmetries, it means that there is an internal relationship between them, which we are not yet able to comprehend! For example, each of the two-dimensional geometries (Euclid, Lobachevsky or Riemann) has its own symmetry group, which acts on the plane. But the points of the plane are complex numbers: in this way the action of any geometric group is transferred to the vast world of complex functions. It is possible and necessary to study the most symmetrical of these functions: AUTOMORPHOUS (which are subject to the Euclid group) and MODULAR (which are subject to the Lobachevsky group)!

There are also elliptic curves in the plane. They have nothing to do with the ellipse, but are given by equations of the form Y 2 = AX 3 + BX 2 + CX and therefore intersect with any straight line at three points. This fact allows us to introduce multiplication among the points of an elliptic curve - to turn it into a group. The algebraic structure of this group reflects the geometric properties of the curve; perhaps it is uniquely determined by its group? This question is worth studying, since for some curves the group of interest to us turns out to be modular, that is, it is related to the Lobachevsky geometry ...

This is how Poincaré reasoned, seducing the mathematical youth of Europe, but at the beginning of the 20th century these temptations did not lead to bright theorems or hypotheses. It turned out differently with Hilbert's call: to study the general solutions of Diophantine equations with integer coefficients! In 1922, the young American Lewis Mordell connected the set of solutions of such an equation (this is a vector space of a certain dimension) with the geometric genus of the complex curve that is given by this equation. Mordell came to the conclusion that if the degree of the equation is sufficiently large (more than two), then the dimension of the solution space is expressed in terms of the genus of the curve, and therefore this dimension is FINITE. On the contrary - to the power of 2, the Pythagorean equation has an INFINITE-DIMENSIONAL family of solutions!

Of course, Mordell saw the connection of his hypothesis with Fermat's theorem. If it becomes known that for every degree n > 2 the space of entire solutions of Fermat's equation is finite-dimensional, this will help to prove that there are no such solutions at all! But Mordell did not see any way to prove his hypothesis - and although he lived a long life, he did not wait for the transformation of this hypothesis into Faltings' theorem. This happened in 1983, in a completely different era, after the great successes of the algebraic topology of manifolds.

Poincaré created this science as if by accident: he wanted to know what three-dimensional manifolds are. After all, Riemann figured out the structure of all closed surfaces and got a very simple answer! If there is no such answer in a three-dimensional or multidimensional case, then you need to come up with a system of algebraic invariants of the manifold that determines its geometric structure. It is best if such invariants are elements of some groups - commutative or non-commutative.

Strange as it may seem, this audacious plan by Poincaré succeeded: it was carried out from 1950 to 1970 thanks to the efforts of a great many geometers and algebraists. Until 1950, there was a quiet accumulation of various methods for classifying manifolds, and after this date, a critical mass of people and ideas seemed to have accumulated and an explosion occurred, comparable to the invention of mathematical analysis in the 17th century. But the analytic revolution lasted for a century and a half, covering the creative biographies of four generations of mathematicians - from Newton and Leibniz to Fourier and Cauchy. On the contrary, the topological revolution of the twentieth century was within twenty years - thanks to a large number its members. At the same time, a large generation of self-confident young mathematicians has emerged, suddenly left without work in their historical homeland.

In the seventies, they rushed into the adjacent areas of mathematics and theoretical physics. Many have created their own scientific schools in dozens of universities in Europe and America. Many students of different ages and nationalities, with different abilities and inclinations, still circulate between these centers, and everyone wants to be famous for some discovery. It was in this pandemonium that Mordell's conjecture and Fermat's theorem were finally proven.

However, the first swallow, unaware of its fate, grew up in Japan in the hungry and unemployed post-war years. The name of the swallow was Yutaka Taniyama. In 1955, this hero turned 28 years old, and he decided (together with friends Goro Shimura and Takauji Tamagawa) to revive mathematical research in Japan. Where to begin? Of course, with overcoming isolation from foreign colleagues! So in 1955, three young Japanese hosted the first international conference on algebra and number theory in Tokyo. It was apparently easier to do this in Japan reeducated by the Americans than in Russia frozen by Stalin ...

Among the guests of honor were two heroes from France: Andre Weil and Jean-Pierre Serre. Here the Japanese were very lucky: Weil was the recognized head of the French algebraists and a member of the Bourbaki group, and the young Serre played a similar role among topologists. In heated discussions with them, the heads of the Japanese youth cracked, their brains melted, but in the end, such ideas and plans crystallized that could hardly have been born in a different environment.

One day, Taniyama approached Weil with a question about elliptic curves and modular functions. At first, the Frenchman did not understand anything: Taniyama was not a master of speaking English. Then the essence of the matter became clear, but Taniyama did not manage to give his hopes an exact formulation. All Weil could reply to the young Japanese was that if he were very lucky in terms of inspiration, then something sensible would grow out of his vague hypotheses. But while the hope for it is weak!

Obviously, Weil did not notice the heavenly fire in Taniyama's gaze. And there was fire: it seems that for a moment the indomitable thought of the late Poincaré moved into the Japanese! Taniyama came to believe that every elliptic curve is generated by modular functions - more precisely, it is "uniformized by a modular form". Alas, this exact wording was born much later - in Taniyama's conversations with his friend Shimura. And then Taniyama committed suicide in a fit of depression... His hypothesis was left without an owner: it was not clear how to prove it or where to test it, and therefore no one took it seriously for a long time. The first response came only thirty years later - almost like in Fermat's era!

The ice broke in 1983, when twenty-seven-year-old German Gerd Faltings announced to the whole world: Mordell's conjecture had been proven! Mathematicians were on their guard, but Faltings was a true German: there were no gaps in his long and complicated proof. It's just that the time has come, facts and concepts have accumulated - and now one talented algebraist, relying on the results of ten other algebraists, has managed to solve a problem that has stood waiting for the master for sixty years. This is not uncommon in 20th-century mathematics. It is worth recalling the secular continuum problem in set theory, Burnside's two conjectures in group theory, or the Poincaré conjecture in topology. Finally, in number theory, the time has come to harvest the old crops ... Which top will be the next in a series of conquered mathematicians? Will Euler's problem, Riemann's hypothesis, or Fermat's theorem collapse? It is good to!

And now, two years after the revelation of Faltings, another inspired mathematician appeared in Germany. His name was Gerhard Frey, and he claimed something strange: that Fermat's theorem is DERIVED from Taniyama's conjecture! Unfortunately, Frey's style of expressing his thoughts was more reminiscent of the unfortunate Taniyama than his clear compatriot Faltings. In Germany, no one understood Frey, and he went overseas - to the glorious town of Princeton, where, after Einstein, they got used to not such visitors. No wonder Barry Mazur, a versatile topologist, one of the heroes of the recent assault on smooth manifolds, made his nest there. And a student grew up next to Mazur - Ken Ribet, equally experienced in the intricacies of topology and algebra, but still not glorifying himself in any way.

When he first heard Frey's speeches, Ribet decided that this was nonsense and near-science fiction (probably, Weil reacted to Taniyama's revelations in the same way). But Ribet could not forget this "fantasy" and at times returned to it mentally. Six months later, Ribet believed that there was something sensible in Frey's fantasies, and a year later he decided that he himself could almost prove Frey's strange hypothesis. But some "holes" remained, and Ribet decided to confess to his boss Mazur. He listened attentively to the student and calmly replied: “Yes, you have done everything! Here you need to apply the transformation Ф, here - use Lemmas B and K, and everything will take on an impeccable form! So Ribet made a leap from obscurity to immortality, using a catapult in the person of Frey and Mazur. In fairness, all of them - along with the late Taniyama - should be considered proofs of Fermat's Last Theorem.

But here's the problem: they derived their statement from the Taniyama hypothesis, which itself has not been proven! What if she's unfaithful? Mathematicians have long known that “anything follows from a lie”, if Taniyama’s guess is wrong, then Ribet’s impeccable reasoning is worthless! We urgently need to prove (or disprove) Taniyama's conjecture - otherwise someone like Faltings will prove Fermat's theorem in a different way. He will become a hero!

It is unlikely that we will ever know how many young or seasoned algebraists jumped on Fermat's theorem after the success of Faltings or after the victory of Ribet in 1986. All of them tried to work in secret, so that in case of failure they would not be ranked among the community of “dummies”-fermatists. It is known that the most successful of all - Andrew Wiles from Cambridge - felt the taste of victory only at the beginning of 1993. This not so much pleased as frightened Wiles: what if his proof of the Taniyama conjecture showed an error or a gap? Then his scientific reputation perished! It is necessary to carefully write down the proof (but it will be many dozens of pages!) And put it off for six months or a year, so that later you can re-read it cold-bloodedly and meticulously ... But what if someone publishes their proof during this time? Oh trouble...

Yet Wiles came up with a double way to quickly test his proof. First, you need to trust one of your reliable friends and colleagues and tell him the whole course of reasoning. From the outside, all the mistakes are more visible! Secondly, it is necessary to read a special course on this topic to smart students and graduate students: these smart people will not miss a single lecturer's mistake! Just do not tell them the ultimate goal of the course until the last moment - otherwise the whole world will know about it! And of course, you need to look for such an audience away from Cambridge - it’s better not even in England, but in America ... What could be better than distant Princeton?

Wiles went there in the spring of 1993. His patient friend Niklas Katz, after listening to Wiles' long report, found a number of gaps in it, but all of them were easily corrected. But the Princeton graduate students soon ran away from Wiles's special course, not wanting to follow the whimsical thought of the lecturer, who leads them to no one knows where. After such a (not particularly deep) review of his work, Wiles decided that it was time to reveal a great miracle to the world.

In June 1993, another conference was held in Cambridge, dedicated to the "Iwasawa theory" - a popular section of number theory. Wiles decided to tell his proof of the Taniyama conjecture on it, without announcing the main result until the very end. The report went on for a long time, but successfully, journalists gradually began to flock, who sensed something. Finally, thunder struck: Fermat's theorem is proved! The general rejoicing was not overshadowed by any doubts: everything seems to be clear ... But two months later, Katz, having read the final text of Wiles, noticed another gap in it. A certain transition in reasoning relied on the "Euler system" - but what Wiles built was not such a system!

Wiles checked the bottleneck and realized that he was mistaken here. Even worse: it is not clear how to replace the erroneous reasoning! This was followed by the darkest months of Wiles' life. Previously, he freely synthesized an unprecedented proof from the material at hand. Now he is tied to a narrow and clear task - without the certainty that it has a solution and that he will be able to find it in the foreseeable future. Recently, Frey could not resist the same struggle - and now his name was obscured by the name of the lucky Ribet, although Frey's guess turned out to be correct. And what will happen to MY guess and MY name?

This hard labor lasted exactly one year. In September 1994, Wiles was ready to admit defeat and leave the Taniyama hypothesis to more fortunate successors. Having made such a decision, he began to slowly reread his proof - from beginning to end, listening to the rhythm of reasoning, re-experiencing the pleasure of successful discoveries. Having reached the "damned" place, Wiles, however, did not mentally hear a false note. Could it be that the course of his reasoning was still flawless, and the error arose only in the VERBAL description of the mental image? If there is no “Euler system” here, then what is hidden here?

Suddenly, a simple thought came to me: the "Euler system" does not work where the Iwasawa theory is applicable. Why not apply this theory directly - fortunately, it is close and familiar to Wiles himself? And why did he not try this approach from the very beginning, but got carried away by someone else's vision of the problem? Wiles could no longer remember these details - and it became useless. He carried out the necessary reasoning within the framework of the Iwasawa theory, and everything turned out in half an hour! Thus - with a delay of one year - the last gap in the proof of Taniyama's conjecture was closed. The final text was given to the mercy of a group of reviewers of the most famous mathematical journal, a year later they declared that now there are no errors. Thus, in 1995, Fermat's last conjecture died at the age of three hundred and sixty, turning into a proven theorem that will inevitably enter the number theory textbooks.

Summing up the three-century fuss around Fermat's theorem, we have to draw a strange conclusion: this heroic epic could not have happened! Indeed, the Pythagorean theorem expresses a simple and important connection between visual natural objects- the length of the segments. But the same cannot be said of Fermat's Theorem. It looks more like a cultural superstructure on a scientific substrate - like reaching the North Pole of the Earth or flying to the moon. Let us recall that both of these feats were sung by writers long before they were accomplished - back in ancient times, after the appearance of Euclid's "Elements", but before the appearance of Diophantus's "Arithmetic". So, then there was a public need for intellectual exploits of this kind - at least imaginary! Previously, the Hellenes had had enough of Homer's poems, just as a hundred years before Fermat, the French had had enough of religious passions. But then religious passions subsided - and science stood next to them.

In Russia, such processes began a hundred and fifty years ago, when Turgenev put Yevgeny Bazarov on a par with Yevgeny Onegin. True, the writer Turgenev poorly understood the motives for the actions of the scientist Bazarov and did not dare to sing them, but this was soon done by the scientist Ivan Sechenov and the enlightened journalist Jules Verne. A spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and here comes first science fiction, and then popular science literature (including the magazine "Knowledge is Power").

At the same time, specific scientific theme not at all important to the general public, and not very important even to the hero performers. So, having heard about the achievement of the North Pole by Peary and Cook, Amundsen instantly changed the goal of his already prepared expedition - and soon reached the South Pole, ahead of Scott by one month. Later, Yuri Gagarin's successful circumnavigation of the Earth forced President Kennedy to change the former goal of the American space program to a more expensive but far more impressive one: landing men on the moon.

Even earlier, the insightful Hilbert answered the naive question of students: “The solution of what scientific problem would be most useful now”? - answered with a joke: “Catch a fly on reverse side Moon! To the perplexed question: “Why is this necessary?” - followed by a clear answer: “Nobody needs THIS! But think about those scientific methods And technical means, which we will have to develop to solve such a problem - and what a lot of other beautiful problems we will solve along the way!

This is exactly what happened with Fermat's Theorem. Euler could well have overlooked it.

In this case, some other problem would become the idol of mathematicians - perhaps also from number theory. For example, the problem of Eratosthenes: is there a finite or infinite set of twin primes (such as 11 and 13, 17 and 19, and so on)? Or Euler's problem: is every even number the sum of two prime numbers? Or: is there an algebraic relation between the numbers π and e? These three problems have not yet been solved, although in the 20th century mathematicians have come close to understanding their essence. But this century has given rise to many new, no less interesting tasks, especially at the intersections of mathematics with physics and other branches of natural science.

Back in 1900, Hilbert singled out one of them: to create a complete system of axioms of mathematical physics! A hundred years later, this problem is far from being solved, if only because the arsenal of mathematical means of physics is steadily growing, and not all of them have a rigorous justification. But after 1970, theoretical physics split into two branches. One (classical) since the time of Newton has been modeling and predicting STABLE processes, the other (newborn) is trying to formalize the interaction of UNSTABLE processes and ways to control them. It is clear that these two branches of physics must be axiomatized separately.

The first of them will probably be dealt with in twenty or fifty years ...

And what is missing from the second branch of physics - the one that is in charge of all kinds of evolution (including outlandish fractals and strange attractors, the ecology of biocenoses and Gumilyov's theory of passionarity)? This we are unlikely to understand soon. But the worship of scientists to the new idol has already become a mass phenomenon. Probably, an epic will unfold here, comparable to the three-century biography of Fermat's theorem. So at the junctions different sciences more and more new idols are born - similar to religious ones, but more complex and dynamic ...

Apparently, a person cannot remain a person without overthrowing the old idols from time to time and without creating new ones - in pain and with joy! Pierre Fermat was lucky to be at a fateful moment close to the hot spot of the birth of a new idol - and he managed to leave an imprint of his personality on the newborn. One can envy such a fate, and it is not a sin to imitate it.

Sergei Smirnov
"Knowledge is power"

HISTORY OF FERMAT'S GREAT THEOREM
A grand affair

Once in the New Year's issue of the mailing list on how to make toasts, I casually mentioned that at the end of the 20th century there was one grandiose event that many did not notice - the so-called Fermat's Last Theorem was finally proved. On this occasion, among the letters I received, I found two responses from girls (one of them, as far as I remember, is Vika, a ninth-grader from Zelenograd), who were surprised by this fact.

And I was surprised by how keenly the girls are interested in the problems of modern mathematics. Therefore, I think that not only girls, but also boys of all ages - from high school students to pensioners, will also be interested in learning the history of the Great Theorem.

The proof of Fermat's theorem is a great event. And since it is not customary to joke with the word "great", then it seems to me that every self-respecting speaker (and all of us, when we say speakers) is simply obliged to know the history of the theorem.

If it so happened that you do not like mathematics as much as I love it, then look at some deepenings in detail with a cursory glance. Understanding that not all readers of our mailing list are interested in wandering in the wilds of mathematics, I tried not to give any formulas (except for the equation of Fermat's theorem and a couple of hypotheses) and to simplify the coverage of some specific issues as much as possible.

How Fermat brewed porridge

The French lawyer and part-time great mathematician of the 17th century, Pierre Fermat (1601-1665), put forward one curious statement from the field of number theory, which later became known as Fermat's Great (or Great) Theorem. This is one of the most famous and phenomenal mathematical theorems. Probably, the excitement around it would not have been so strong if in the book of Diophantus of Alexandria (3rd century AD) "Arithmetic", which Fermat often studied, making notes on its wide margins, and which his son Samuel kindly preserved for posterity , approximately the following entry of the great mathematician was not found:

"I have a very startling piece of evidence, but it's too big to fit in the margins."

It was this entry that caused the subsequent grandiose turmoil around the theorem.

So, the famous scientist said that he had proved his theorem. Let's ask ourselves the question: did he really prove it or did he lie corny? Or are there other versions explaining the appearance of that marginal entry that did not allow many mathematicians of the next generations to sleep peacefully?

The history of the Great Theorem is as fascinating as an adventure through time. Fermat stated in 1636 that an equation of the form x n + y n =z n has no solutions in integers with exponent n>2. This is actually Fermat's Last Theorem. In this seemingly simple mathematical formula, the Universe has masked incredible complexity. The Scottish-born American mathematician Eric Temple Bell, in his book The Final Problem (1961), even suggested that perhaps humanity would cease to exist before it could prove Fermat's Last Theorem.

It is somewhat strange that for some reason the theorem was late with its birth, since the situation was long overdue, because its special case for n = 2 - another famous mathematical formula - the Pythagorean theorem, arose twenty-two centuries earlier. Unlike Fermat's theorem, the Pythagorean theorem has an infinite number of integer solutions, for example, such Pythagorean triangles: (3,4,5), (5,12,13), (7,24,25), (8,15,17 ) … (27,36,45) … (112,384,400) … (4232, 7935, 8993) …

Grand Theorem Syndrome

Who just did not try to prove Fermat's theorem. Any fledgling student considered it his duty to apply to the Great Theorem, but no one was able to prove it. At first it didn't work for a hundred years. Then a hundred more. And further. A mass syndrome began to develop among mathematicians: "How is it? Fermat proved it, but what if I can't, or what?" - and some of them on this basis went crazy in full sense this word.

No matter how much the theorem was tested, it always turned out to be true. I knew one energetic programmer who was obsessed with the idea of ​​disproving the Great Theorem by trying to find at least one of its solutions (counterexample) by iterating over integers using a fast computer (at that time more commonly called a computer). He believed in the success of his enterprise and liked to say: "A little more - and a sensation will break out!" I think that in different parts of our planet there were a considerable number of this kind of bold seekers. Of course, he did not find any solution. And no computers, even with fabulous speed, could ever check the theorem, because all the variables of this equation (including the exponents) can increase to infinity.

Theorem requires proof

Mathematicians know that if a theorem is not proven, anything (either true or false) can follow from it, as it did with some other hypotheses. For example, in one of his letters, Pierre Fermat suggested that numbers of the form 2 n +1 (the so-called Fermat numbers) are necessarily prime (that is, they do not have integer divisors and are only divisible without remainder by themselves and by one), if n is a power of two (1, 2, 4, 8, 16, 32, 64, etc.). Fermat's hypothesis lived for more than a hundred years - until Leonhard Euler showed in 1732 that

2 32 +1 = 4 294 967 297 = 6 700 417 641

Then, almost 150 years later (1880), Fortune Landry factored the following Fermat number:

2 64 +1 = 18 446 744 073 709 551 617 = 274 177 67 280 421 310 721

How they could find the divisors of these large numbers without the help of computers - God only knows. In turn, Euler put forward the hypothesis that the equation x 4 + y 4 + z 4 =u 4 has no solutions in integers. However, about 250 years later, in 1988, Nahum Elkis from Harvard managed to discover (already with the help of computer program), What

2 682 440 4 + 15 365 639 4 + 18 796 760 4 = 20 615 673 4

Therefore, Fermat's Last Theorem required proof, otherwise it was just a hypothesis, and it could well be that somewhere in the endless numerical fields the solution to the equation of the Great Theorem was lost.

The most virtuoso and prolific mathematician of the 18th century, Leonhard Euler, whose archive of records mankind has been sorting out for almost a century, proved Fermat's theorem for powers 3 and 4 (or rather, he repeated the lost proofs of Pierre Fermat himself); his follower in number theory, Legendre (and independently Dirichlet) - for degree 5; Lame - for degree 7. But in general view the theorem remained unproved.

On March 1, 1847, at a meeting of the Paris Academy of Sciences, two eminent mathematician- Gabriel Lame and Augustin Cauchy - said they had come to the end of the proof of the Great Theorem and had a race, publishing their proofs in parts. However, the duel between them was interrupted because the same error was discovered in their proofs, which was pointed out by the German mathematician Ernst Kummer.

At the beginning of the 20th century (1908), a wealthy German entrepreneur, philanthropist and scientist Paul Wolfskel bequeathed one hundred thousand marks to anyone who would present a complete proof of Fermat's theorem. Already in the first year after the publication of Wolfskell's testament by the Göttingen Academy of Sciences, it was inundated with thousands of proofs from lovers of mathematics, and this flow did not stop for decades, but, as you can imagine, they all contained errors. They say that the academy prepared forms with the following content:

Dear __________________________!
In your proof of Fermat's Theorem on ____ page ____ line from the top
The following error was found in the formula:__________________________:,

Which were sent to unlucky applicants for the award.

At that time, a semi-contemptuous nickname appeared in the circle of mathematicians - fermist. This was the name of any self-confident upstart who lacked knowledge, but more than had ambitions to hastily try his hand at proving the Great Theorem, and then, not noticing his own mistakes, proudly slapping his chest, loudly declare: "I proved the first Fermat's Theorem! Every farmer, even if he was ten thousandth in number, considered himself the first - this was ridiculous. Simple appearance The Great Theorem reminded Fermists so much of an easy prey that they were absolutely not embarrassed that even Euler and Gauss could not cope with it.

(Fermists, oddly enough, still exist today. Although one of them did not believe that he had proved the theorem like a classical fermist, but until recently he made attempts - he refused to believe me when I told him that Fermat's theorem had already been proved).

The most powerful mathematicians, perhaps in the quiet of their offices, also tried to cautiously approach this unbearable rod, but did not talk about it aloud, so as not to be branded as Fermists and, thus, not to harm their high authority.

By that time, the proof of the theorem for the exponent n appeared<100. Потом для n<619. Надо ли говорить о том, что все доказательства невероятно сложны. Но в общем виде теорема оставалась недоказанной.

Strange hypothesis

Until the middle of the twentieth century, no major advances in the history of the Great Theorem were observed. But soon an interesting event took place in mathematical life. In 1955, the 28-year-old Japanese mathematician Yutaka Taniyama advanced a statement from a completely different area of ​​mathematics, called the Taniyama Hypothesis (aka the Taniyama-Shimura-Weil Hypothesis), which, unlike Fermat's belated Theorem, was ahead of its time.

Taniyama's conjecture states: "to every elliptic curve there corresponds a certain modular form." This statement for mathematicians of that time sounded about as absurd as the statement sounds for us: "a certain metal corresponds to each tree." It is easy to guess how a normal person can relate to such a statement - he simply will not take it seriously, which happened: mathematicians unanimously ignored the hypothesis.

A little explanation. Elliptic curves, known for a long time, have a two-dimensional form (located on a plane). Modular functions, discovered in the 19th century, have a four-dimensional form, so we cannot even imagine them with our three-dimensional brains, but we can describe them mathematically; in addition, modular forms are amazing in that they have the utmost possible symmetry - they can be translated (shifted) in any direction, mirrored, fragments can be swapped, rotated in infinitely many ways - and their appearance does not change. As you can see, elliptic curves and modular forms have little in common. Taniyama's hypothesis states that the descriptive equations of these two absolutely different mathematical objects corresponding to each other can be expanded into the same mathematical series.

Taniyama's hypothesis was too paradoxical: it combined completely different concepts - rather simple flat curves and unimaginable four-dimensional shapes. This never occurred to anyone. When, at an international mathematical symposium in Tokyo in September 1955, Taniyama demonstrated several correspondences between elliptic curves and modular forms, everyone saw this as nothing more than a funny coincidence. To Taniyama's modest question: is it possible to find the corresponding modular function for each elliptic curve, the venerable Frenchman Andre Weil, who at that time was one of the world's best specialists in number theory, gave a quite diplomatic answer, what, they say, if the inquisitive Taniyama does not leave enthusiasm, then maybe he will be lucky and his incredible hypothesis will be confirmed, but this must not happen soon. In general, like many other outstanding discoveries, at first Taniyama's hypothesis was ignored, because they had not grown up to it yet - almost no one understood it. Only one colleague of Taniyama, Goro Shimura, knowing his highly gifted friend well, intuitively felt that his hypothesis was correct.

Three years later (1958), Yutaka Taniyama committed suicide (however, samurai traditions are strong in Japan). From the point of view of common sense - an incomprehensible act, especially when you consider that very soon he was going to get married. The leader of young Japanese mathematicians began his suicide note as follows: “Yesterday I did not think about suicide. Recently, I often heard from others that I was tired mentally and physically. Actually, even now I don’t understand why I’m doing this ...” and so on on three sheets. It’s a pity, of course, that this was the fate of an interesting person, but all geniuses are a little strange - that’s why they are geniuses (for some reason, the words of Arthur Schopenhauer came to mind: “in ordinary life, a genius is as much use as a telescope in a theater”) . The hypothesis has been abandoned. Nobody knew how to prove it.

For ten years, Taniyama's hypothesis was hardly mentioned. But in the early 70s, it became popular - it was regularly checked by everyone who could understand it - and it was always confirmed (as, in fact, Fermat's theorem), but, as before, no one could prove it.

The amazing connection between the two hypotheses

Another 15 years have passed. In 1984, there was one key event in the life of mathematics that combined the extravagant Japanese conjecture with Fermat's Last Theorem. The German Gerhard Frey put forward a curious statement, similar to a theorem: "If Taniyama's conjecture is proved, then, consequently, Fermat's Last Theorem will be proved." In other words, Fermat's theorem is a consequence of Taniyama's conjecture. (Frey, using ingenious mathematical transformations, reduced Fermat's equation to the form of an elliptic curve equation (the same one that appears in Taniyama's hypothesis), more or less substantiated his assumption, but could not prove it). And just a year and a half later (1986), a professor at the University of California, Kenneth Ribet, clearly proved Frey's theorem.

What happened now? Now it turned out that, since Fermat's theorem is already exactly a consequence of Taniyama's conjecture, all that is needed is to prove the latter in order to break the laurels of the conqueror of the legendary Fermat's theorem. But the hypothesis turned out to be difficult. In addition, over the centuries, mathematicians became allergic to Fermat's theorem, and many of them decided that it would also be almost impossible to cope with Taniyama's conjecture.

The death of Fermat's hypothesis. The birth of a theorem

Another 8 years have passed. One progressive English professor of mathematics from Princeton University (New Jersey, USA), Andrew Wiles, thought he had found a proof of Taniyama's conjecture. If the genius is not bald, then, as a rule, disheveled. Wiles is disheveled, therefore, looks like a genius. Entering into History, of course, is tempting and very desirable, but Wiles, like a real scientist, did not flatter himself, realizing that thousands of Fermists before him also saw ghostly evidence. Therefore, before presenting his proof to the world, he carefully checked it himself, but realizing that he could have a subjective bias, he also involved others in the checks, for example, under the guise of ordinary mathematical tasks, he sometimes threw various fragments of his proof to smart graduate students. Wiles later admitted that no one but his wife knew he was working on proving the Great Theorem.

And so, after long checks and painful reflections, Wiles finally plucked up courage, and perhaps, as he himself thought, arrogance, and on June 23, 1993, at a mathematical conference on number theory in Cambridge, he announced his great achievement.

It was, of course, a sensation. No one expected such agility from a little-known mathematician. Then the press came along. Everyone was tormented by a burning interest. Slender formulas, like the strokes of a beautiful picture, appeared before the curious eyes of the audience. Real mathematicians, after all, they are like that - they look at all sorts of equations and see in them not numbers, constants and variables, but they hear music, like Mozart looking at a musical staff. Just like when we read a book, we look at the letters, but we don’t seem to notice them, but immediately perceive the meaning of the text.

The presentation of the proof seemed to be successful - no errors were found in it - no one heard a single false note (although most mathematicians simply stared at him like first-graders at an integral and did not understand anything). Everyone decided that a large-scale event had happened: Taniyama's hypothesis was proved, and consequently Fermat's Last Theorem. But about two months later, a few days before the manuscript of Wiles's proof was to go into circulation, it was found to be inconsistent (Katz, a colleague of Wiles, noted that one piece of reasoning relied on "Euler's system", but what built by Wiles, was not such a system), although, in general, Wiles's techniques were considered interesting, elegant and innovative.

Wiles analyzed the situation and decided that he had lost. One can imagine how he felt with all his being what it means "from the great to the ridiculous one step." "I wanted to enter History, but instead I joined a team of clowns and comedians - arrogant farmists" - approximately such thoughts exhausted him during that painful period of his life. For him, a serious mathematician, it was a tragedy, and he threw his proof on the back burner.

But a little over a year later, in September 1994, while thinking about that bottleneck of the proof together with his colleague Taylor from Oxford, the latter suddenly had the idea that the "Euler system" could be changed to the Iwasawa theory (section of number theory). Then they tried to use the Iwasawa theory, doing without the "Euler system", and they all came together. The corrected version of the proof was submitted for verification, and a year later it was announced that everything in it was absolutely clear, without a single mistake. In the summer of 1995, in one of the leading mathematical journals - "Annals of Mathematics" - a complete proof of Taniyama's conjecture (hence, Fermat's Great (Large) Theorem) was published, which occupied the entire issue - over one hundred pages. The proof is so complex that only a few dozen people around the world could understand it in its entirety.

Thus, at the end of the 20th century, the whole world recognized that in the 360th year of its life, Fermat's Last Theorem, which in fact had been a hypothesis all this time, had become a proven theorem. Andrew Wiles proved Fermat's Great (Great) Theorem and entered History.

Think you've proven a theorem...

The happiness of the discoverer always goes to someone alone - it is he who, with the last blow of the hammer, cracks the hard nut of knowledge. But one cannot ignore the many previous blows that have formed a crack in the Great Theorem for centuries: Euler and Gauss (the kings of mathematics of their time), Evariste Galois (who managed to establish the theory of groups and fields in his short 21-year life, whose works were recognized as brilliant only after his death), Henri Poincaré (the founder of not only bizarre modular forms, but also conventionalism - a philosophical trend), David Gilbert (one of the strongest mathematicians of the twentieth century), Yutaku Taniyama, Goro Shimura, Mordell, Faltings, Ernst Kummer, Barry Mazur, Gerhard Frey, Ken Ribbet, Richard Taylor and others real scientists(I'm not afraid of these words).

The proof of Fermat's Last Theorem can be put on a par with such achievements of the twentieth century as the invention of the computer, the nuclear bomb, and space flight. Although not so widely known about it, because it does not invade the zone of our momentary interests, such as a TV or an electric light bulb, but it was a flash of a supernova, which, like all immutable truths, will always shine on humanity.

You can say: "Just think, you proved some kind of theorem, who needs it?". A fair question. David Gilbert's answer will fit exactly here. When, to the question: "what is the most important task for science now?", He answered: "to catch a fly on the far side of the moon", he was reasonably asked: "but who needs it?", he replied like this:" Nobody needs it. But think about how important the most difficult tasks think about how many problems mankind has been able to solve in 360 years before proving Fermat's theorem. In search of its proof, almost half of modern mathematics was discovered. We must also take into account that mathematics is the avant-garde of science (and , by the way, the only one of the sciences that is built without a single mistake), and any scientific achievements and inventions begin right here. As Leonardo da Vinci noted, "only that doctrine that is confirmed mathematically can be recognized as a science."

* * *

And now let's go back to the beginning of our story, remember Pierre Fermat's entry in the margins of Diophantus's textbook and once again ask ourselves: did Fermat really prove his theorem? Of course, we cannot know this for sure, and as in any case, different versions arise here:

Version 1: Fermat proved his theorem. (To the question: "Did Fermat have exactly the same proof of his theorem?", Andrew Wiles remarked: "Fermat could not have so proof. This is a proof of the 20th century. "We understand that in the 17th century mathematics, of course, was not the same as at the end of the 20th century - in that era, d, Artagnan, the queen of sciences, did not yet possess those discoveries (modular forms, Taniyama's theorems , Frey, etc.), which only made it possible to prove Fermat's Last Theorem. Of course, one can assume: what the hell is not joking - what if Fermat guessed in a different way? This version, although probable, is practically impossible according to most mathematicians);
Version 2: It seemed to Pierre de Fermat that he had proved his theorem, but there were errors in his proof. (That is, Fermat himself was also the first Fermatist);
Version 3: Fermat did not prove his theorem, but simply lied in the margins.

If one of the last two versions is correct, which is most likely, then a simple conclusion can be drawn: great people, although they are great, they can also make mistakes or sometimes do not mind lying(basically, this conclusion will be useful for those who are inclined to completely trust their idols and other rulers of thoughts). Therefore, when reading the works of authoritative sons of mankind or listening to their pathetic speeches, you have every right to doubt their statements. (Please note that to doubt is not to reject).

Reprinting of article materials is possible only with obligatory links to the site (on the Internet - hyperlink) and to the author

FERMAT GREAT THEOREM - the statement of Pierre Fermat (a French lawyer and part-time mathematician) that the Diophantine equation X n + Y n = Z n , with an exponent n>2, where n = an integer, has no solutions in positive integers . Author's text: "It is impossible to decompose a cube into two cubes, or a bi-square into two bi-squares, or in general a power greater than two into two powers with the same exponent."

"Fermat and his theorem", Amadeo Modigliani, 1920

Pierre came up with this theorem on March 29, 1636. And after some 29 years, he died. But that's where it all started. After all, a wealthy German mathematician by the name of Wolfskel bequeathed one hundred thousand marks to the one who presents the complete proof of Fermat's theorem! But the excitement around the theorem was connected not only with this, but also with professional mathematical excitement. Fermat himself hinted to the mathematical community that he knew the proof - shortly before his death, in 1665, he left the following entry in the margins of the book Diophantus of Alexandria "Arithmetic": "I have a very amazing proof, but it is too large to be placed on fields."

It was this hint (plus, of course, a cash prize) that made mathematicians unsuccessfully spend their best years(According to the calculations of American scientists, only professional mathematicians spent 543 years on this in total).

At some point (in 1901), work on Fermat's theorem acquired the dubious fame of "work akin to the search for a perpetual motion machine" (there was even a derogatory term - "fermatists"). And suddenly, on June 23, 1993, at a mathematical conference on number theory in Cambridge, English professor of mathematics from Princeton University (New Jersey, USA) Andrew Wiles announced that he had finally proved Fermat!

The proof, however, was not only complicated, but also obviously erroneous, as Wiles was pointed out by his colleagues. But Professor Wiles dreamed of proving the theorem all his life, so it is not surprising that in May 1994 he presented a new, improved version of the proof to the scientific community. There was no harmony, beauty in it, and it was still very complicated - the fact that mathematicians have been analyzing this proof for a whole year (!) To understand whether it is not erroneous, speaks for itself!

But in the end, Wiles' proof was found to be correct. But mathematicians did not forgive Pierre Fermat for his very hint in Arithmetic, and, in fact, they began to consider him a liar. In fact, the first person to question Fermat's moral integrity was Andrew Wiles himself, who remarked that "Fermat could not have had such proof. This is twentieth-century proof." Then, among other scientists, the opinion became stronger that Fermat "could not prove his theorem in another way, and Fermat could not prove it in the way that Wiles went, for objective reasons."

In fact, Fermat, of course, could prove it, and a little later this proof will be recreated by the analysts of the New Analytical Encyclopedia. But - what are these "objective reasons"?
In fact, there is only one such reason: in those years when Fermat lived, Taniyama’s conjecture could not appear, on which Andrew Wiles built his proof, because the modular functions that Taniyama’s conjecture operates on were discovered only in late XIX century.

How did Wiles himself prove the theorem? The question is not idle - this is important for understanding how Fermat himself could prove his theorem. Wiles built his proof on the proof of Taniyama's conjecture put forward in 1955 by the 28-year-old Japanese mathematician Yutaka Taniyama.

The conjecture sounds like this: "every elliptic curve corresponds to a certain modular form." Elliptic curves, known for a long time, have a two-dimensional form (located on a plane), while modular functions have a four-dimensional form. That is, Taniyama's hypothesis combined completely different concepts - simple flat curves and unimaginable four-dimensional forms. The very fact of connecting different-dimensional figures in the hypothesis seemed absurd to scientists, which is why in 1955 it was not given any importance.

However, in the fall of 1984, the "Taniyama hypothesis" was suddenly remembered again, and not only remembered, but its possible proof was connected with the proof of Fermat's theorem! This was done by Saarbrücken mathematician Gerhard Frey, who told the scientific community that "if anyone could prove Taniyama's conjecture, then Fermat's Last Theorem would be proved."

What did Frey do? He converted Fermat's equation to a cubic one, then drew attention to the fact that an elliptic curve obtained by converting Fermat's equation to a cubic one cannot be modular. However, Taniyama's conjecture stated that any elliptic curve could be modular! Accordingly, an elliptic curve constructed from Fermat's equation cannot exist, which means there cannot be entire solutions and Fermat's theorem, which means it is true. Well, in 1993, Andrew Wiles simply proved Taniyama's conjecture, and hence Fermat's theorem.

However, Fermat's theorem can be proved much more simply, on the basis of the same multidimensionality that both Taniyama and Frey operated on.

To begin with, let's pay attention to the condition stipulated by Pierre Fermat himself - n>2. Why was this condition necessary? Yes, only for the fact that for n=2 the ordinary Pythagorean theorem X 2 +Y 2 =Z 2 becomes a special case of Fermat's theorem, which has an infinite number of integer solutions - 3,4,5; 5,12,13; 7.24.25; 8,15,17; 12,16,20; 51,140,149 and so on. Thus, the Pythagorean theorem is an exception to Fermat's theorem.

But why exactly in the case of n=2 does such an exception occur? Everything falls into place if you see the relationship between the degree (n=2) and the dimension of the figure itself. The Pythagorean triangle is a two-dimensional figure. Not surprisingly, Z (that is, the hypotenuse) can be expressed in terms of legs (X and Y), which can be integers. The size of the angle (90) makes it possible to consider the hypotenuse as a vector, and the legs are vectors located on the axes and coming from the origin. Accordingly, it is possible to express a two-dimensional vector that does not lie on any of the axes in terms of the vectors that lie on them.

Now, if we go to the third dimension, and hence to n=3, in order to express a three-dimensional vector, there will not be enough information about two vectors, and therefore it will be possible to express Z in Fermat's equation in at least three terms (three vectors lying, respectively, on the three axes of the coordinate system).

If n=4, then there should be 4 terms, if n=5, then there should be 5 terms, and so on. In this case, there will be more than enough whole solutions. For example, 3 3 +4 3 +5 3 =6 3 and so on (you can choose other examples for n=3, n=4 and so on).

What follows from all this? It follows from this that Fermat's theorem does indeed have no entire solutions for n>2 - but only because the equation itself is incorrect! With the same success, one could try to express the volume of a parallelepiped in terms of the lengths of its two edges - of course, this is impossible (whole solutions will never be found), but only because to find the volume of a parallelepiped, you need to know the lengths of all three of its edges.

When the famous mathematician David Gilbert was asked what is the most important task for science now, he answered "to catch a fly on the far side of the moon." To the reasonable question "Who needs it?" he answered like this: "No one needs it. But think about how many important and complex tasks you need to solve in order to accomplish this."

In other words, Fermat (a lawyer first of all!) played a witty legal joke on the entire mathematical world based on wrong staging tasks. He, in fact, suggested that mathematicians find an answer why a fly cannot live on the other side of the Moon, and in the margins of Arithmetic he only wanted to write that there is simply no air on the Moon, i.e. there can be no integer solutions of his theorem for n>2 only because each value of n must correspond to a certain number of terms on the left side of his equation.

But was it just a joke? Not at all. Fermat's genius lies precisely in the fact that he was actually the first to see the relationship between the degree and the dimension of a mathematical figure - that is, which is absolutely equivalent, the number of terms on the left side of the equation. The meaning of his famous theorem was precisely to not only push the mathematical world on the idea of ​​this relationship, but also to initiate the proof of the existence of this relationship - intuitively understandable, but mathematically not yet substantiated.

Fermat, like no one else, understood that establishing a relationship between seemingly different objects is extremely fruitful not only in mathematics, but also in any science. Such a relationship points to some deep principle underlying both objects and allowing a deeper understanding of them.

For example, initially physicists considered electricity and magnetism as completely unrelated phenomena, and in the 19th century, theorists and experimenters realized that electricity and magnetism were closely related. The result was a deeper understanding of both electricity and magnetism. Electric currents generate magnetic fields, and magnets can induce electricity in conductors near the magnets. This led to the invention of dynamos and electric motors. Eventually it was discovered that light is the result of concerted harmonic vibrations magnetic and electric fields.

The mathematics of Fermat's time consisted of islands of knowledge in a sea of ​​ignorance. Geometers studied shapes on one island, and mathematicians studied probability and chance on the other island. The language of geometry was very different from the language of probability theory, and algebraic terminology was alien to those who spoke only about statistics. Unfortunately, mathematics of our time consists of approximately the same islands.

Farm was the first to realize that all these islands are interconnected. And his famous theorem - Fermat's GREAT THEOREM - is an excellent confirmation of this.

So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in its essence and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n> 2. Everything seems to be simple and clear, but the best mathematicians and ordinary amateurs fought over searching for a solution for more than three and a half centuries.

Why is she so famous? Now let's find out...



Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with 5th grade high school, but the proof is not even any professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved for so long. 2. What does it consist of?

Let's start with Pythagorean pants The wording is really simple - at first glance. As we know from childhood, "Pythagorean pants are equal on all sides." The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle the square built on the hypotenuse is equal to the sum of the squares built on the legs.

In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triples satisfying the equation x²+y²=z². They proved that Pythagorean triplets infinitely many, and obtained general formulas for finding them. They must have tried looking for threes or more. high degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.


That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²

Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.

Or 5, 12, 13: 25 + 144 = 169. Great.

Well, and so on. What if we take a similar equation x³+y³=z³ ? Maybe there are such numbers too?




And so on (Fig. 1).

Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.

It is more difficult to prove the absence: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give a solution). And that's it, the opponent is defeated. How to prove absence?

To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.

In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2):


And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:





But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically studied the general equation x n+yn=zn . And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! All that remains is his remark in Diophantus' Arithmetic: "I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it."

Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.

After Fermat, great minds such as Leonhard Euler worked on finding the proof (in 1770 he proposed a solution for n = 3),

Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat's Last Theorem, but only in 1993 did mathematicians see and believe that the three-century saga of finding a proof of Fermat's last theorem was almost over.

It is easy to show that it suffices to prove Fermat's theorem only for prime n: 3, 5, 7, 11, 13, 17, … For composite n, the proof remains valid. But there are infinitely many prime numbers...

In 1825, using the method of Sophie Germain, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.


Finally, the German mathematician Ernst Kummer showed in a brilliant study that the methods of mathematics in the 19th century cannot prove the theorem in general form. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.

In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.

He soon died of natural causes. The heirs were pretty surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...

Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:


Dear (s). . . . . . . .

Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau











In 1963, Paul Cohen, drawing on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems, the continuum hypothesis. What if Fermat's Last Theorem is also unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups of programmers and mathematicians proved Fermat's Last Theorem for all values ​​of n up to 500, then up to 1,000, and later up to 10,000.

In the 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians claimed that Fermat's Last Theorem was true for all values ​​of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.




In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.

Nevertheless, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole building could collapse at any moment.

In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proved that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proved. But for thirty years it was not possible to prove the Taniyama–Shimura conjecture, and there were less and less hopes for success.

In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.

Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama–Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat’s Last Theorem is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.

In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.







While the hype continued in the press, serious work began to verify the evidence. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.

It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.

“...half a minute after the start of the festive dinner on the occasion of her birthday, I gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?






This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.

A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!

Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ...