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 my 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 the 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 of the ancient world of numbers, the laws of which Pythagoras, Euclid and Archimedes barely had time to feel. Note that Diophantus did not yet master the positional system of writing large numbers, but he knew what negative numbers were 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 with the quadratic Pythagorean equation, 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?
Quite simply: by the middle of the 17th century, arithmetic had gone 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 the complete triumph of 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, complex numbers appeared that 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, that 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 ALL prime degrees (!) - it seemed to him in a rush, 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 pejorative term 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 hint was clear: follow the example of Gauss and Galois, create general theories of 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 20th century was within twenty years, thanks to the large number of its participants. 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 fields 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?
That's where Wiles went 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. Was the course of his reasoning still impeccable, 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 lengths of 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, a specific scientific topic is not at all important for the general public and is not very important even for the heroes-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 the far side of the moon!” To the perplexed question: “Why is this necessary?” - followed by a clear answer: “Nobody needs THIS! But think of the scientific methods and technical means that 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 also gave rise to many new, no less interesting problems, especially at the intersection 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. Thus, at the intersection of different sciences, 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"
There are not many people in the world who have never heard of Fermat's Last Theorem- perhaps this is the only mathematical problem that has received such wide popularity and has become a real legend. It is mentioned in many books and films, while the main context of almost all mentions is impossibility to prove a theorem.
Yes, this theorem is very famous and in a sense has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this happened back in 1995. But first things first.
So, Fermat's Last Theorem (often referred to as Fermat's Last Theorem), formulated in 1637 by a 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 n + b n \u003d c n has no natural (that is, non-fractional) solutions for n > 2. Everything seems to be simple and clear, but the best mathematicians and simple amateurs have been struggling to find a solution for more than three and a half centuries.
Fermat himself claimed to have derived a very simple and concise proof of his theory, but so far no documentary evidence of this fact has been found. Therefore, it is now believed that Fermat was never able to find a general solution to his theorem., although he wrote a partial proof for n = 4.
After Fermat, such great minds as Leonhard Euler(in 1770 he proposed a solution for n = 3), Adrien Legendre and Johann Dirichlet(these scientists jointly found evidence 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 a final solution
Fermat's Last Theorem, but it wasn't until 1993 that mathematicians saw and believed that the three-century saga of finding a proof of Fermat's Last Theorem was almost over.
In 1993, an English mathematician Andrew Wiles presented to the world proof of Fermat's Last Theorem which has been in the works for more than seven years. But 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.
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 ...
Grigory Perelman. Refusenik
Vasily Maksimov
In August 2006, the names of the world's best mathematicians were announced, who received the most prestigious Fields Medal - a kind of analogue of the Nobel Prize, which mathematicians, at the whim of Alfred Nobel, were deprived of. The Fields Medal - in addition to the badge of honor, laureates are awarded a check for fifteen thousand Canadian dollars - is awarded by the International Congress of Mathematicians every four years. It was established by Canadian scientist John Charles Fields and was first awarded in 1936. Since 1950, the Fields Medal has been awarded regularly personally by the King of Spain for his contribution to the development of mathematical science. From one to four scientists under the age of forty can become laureates of the award. Forty-four mathematicians have already received the prize, including eight Russians.
Grigory Perelman. Henri Poincare.
In 2006, the Frenchman Wendelin Werner, the Australian Terence Tao and two Russians, Andrey Okounkov, who works in the USA, and Grigory Perelman, a scientist from St. Petersburg, became laureates. However, at the last moment it became known that Perelman refused this prestigious award - as the organizers announced, "for reasons of principle."
Such an extravagant act of the Russian mathematician did not come as a surprise to people who knew him. This is not the first time he refuses mathematical awards, explaining his decision by the fact that he does not like solemn events and excessive hype around his name. Ten years ago, in 1996, Perelman refused the prize of the European Mathematical Congress, citing the fact that he had not finished work on the scientific problem nominated for the award, and this was not the last case. The Russian mathematician seems to have made it his life's goal to surprise people, going against public opinion and the scientific community.
Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad. From a young age, he was fond of the exact sciences, brilliantly graduated from the famous 239th secondary school with in-depth study of mathematics, won numerous mathematical competitions: for example, in 1982, as part of a team of Soviet schoolchildren, he participated in the International Mathematical Olympiad, held in Budapest. Perelman without exams was enrolled in the mechanics and mathematics department of Leningrad University, where he studied "excellently", continuing to win in mathematical competitions at all levels. After graduating from the university with honors, he entered graduate school at the St. Petersburg Department of the Steklov Mathematical Institute. His supervisor was the famous mathematician Academician Alexandrov. Having defended his Ph.D. thesis, Grigory Perelman remained at the institute, in the laboratory of geometry and topology. Known for his work on the theory of Alexandrov spaces, he was able to find evidence for a number of important hypotheses. Despite numerous offers from leading Western universities, Perelman prefers to work in Russia.
His most notorious success was the solution in 2002 of the famous Poincare conjecture, published in 1904 and since then remained unproven. Perelman worked on it for eight years. The Poincaré hypothesis was considered one of the greatest mathematical mysteries, and its solution was considered the most important achievement in mathematical science: it would instantly advance the study of the problems of the physical and mathematical foundations of the universe. The brightest minds on the planet predicted its solution only in a few decades, and the Clay Institute of Mathematics in Cambridge, Massachusetts, made the Poincare problem one of the seven most interesting unsolved mathematical problems of the millennium, each of which was promised a million dollar prize (Millennium Prize Problems) .
The hypothesis (sometimes called the problem) of the French mathematician Henri Poincaré (1854–1912) is formulated as follows: any closed, simply connected three-dimensional space is homeomorphic to a three-dimensional sphere. For clarification, a good example is used: if you wrap an apple with a rubber band, then, in principle, by pulling the tape together, you can squeeze the apple into a point. If you wrap a donut with the same tape, then you cannot squeeze it into a point without tearing either the donut or rubber. In this context, an apple is called a "singly connected" figure, but a donut is not simply connected. Almost a hundred years ago, Poincaré established that the two-dimensional sphere is simply connected and suggested that the three-dimensional sphere is also simply connected. The best mathematicians in the world could not prove this conjecture.
To qualify for the Clay Institute prize, Perelman only had to publish his solution in one of the scientific journals, and if within two years no one can find an error in his calculations, then the solution will be considered correct. However, Perelman deviated from the rules from the very beginning, publishing his solution on the preprint site of the Los Alamos Science Laboratory. Perhaps he was afraid that an error had crept into his calculations - a similar story had already happened in mathematics. In 1994, the English mathematician Andrew Wiles proposed a solution to the famous Fermat's theorem, and a few months later it turned out that an error had crept into his calculations (although it was later corrected, and the sensation still took place). There is still no official publication of the proof of the Poincare conjecture - but there is an authoritative opinion of the best mathematicians on the planet, confirming the correctness of Perelman's calculations.
The Fields Medal was awarded to Grigory Perelman precisely for solving the Poincaré problem. But the Russian scientist refused the prize, which he undoubtedly deserves. “Grigory told me that he feels isolated from the international mathematical community, outside of this community, and therefore does not want to receive an award,” John Ball, president of the World Union of Mathematicians (WCM), said at a press conference in Madrid.
There are rumors that Grigory Perelman is going to leave science altogether: six months ago he quit his native Steklov Mathematical Institute, and they say that he will no longer do mathematics. Perhaps the Russian scientist believes that by proving the famous hypothesis, he has done everything he could for science. But who will undertake to talk about the train of thought of such a bright scientist and extraordinary person? .. Perelman refuses any comments, and he told The Daily Telegraph newspaper: “Nothing that I can say is of the slightest public interest.” However, the leading scientific publications were unanimous in their assessments when they reported that "Grigory Perelman, having solved the Poincare theorem, stood on a par with the greatest geniuses of the past and present."
Monthly literary and journalistic magazine and publishing house.
Many years ago, I received a letter from Tashkent from Valery Muratov, judging by the handwriting, a man of youthful age, who then lived on Kommunisticheskaya Street in the house number 31. The guy was determined: “Directly to the point. How much will you pay me for proving Fermat’s theorem? suits at least 500 rubles. At another time, I would have proved it to you for free, but now I need money ... "
An amazing paradox: few people know who Fermat is, when he lived and what he did. Even fewer people can even describe his great theorem in the most general terms. But everyone knows that there is some kind of Fermat's theorem, over the proof of which mathematicians of the whole world have been struggling for more than 300 years, but they cannot prove it!
There are many ambitious people, and the very consciousness that there is something that others cannot do, further spurs their ambition. Therefore, thousands (!) of proofs of the Great Theorem have come and come to academies, scientific institutes, and even newspaper editorial offices around the world - an unprecedented and never broken record of pseudoscientific amateur performance. There is even a term: "fermatists", that is, people obsessed with the desire to prove the Great Theorem, who completely exhausted professional mathematicians with demands to evaluate their work. The famous German mathematician Edmund Landau even prepared a standard, according to which he answered: "There is an error on the page in your proof of Fermat's theorem ...", and his graduate students put down the page number. And in the summer of 1994, newspapers around the world report something completely sensational: The Great Theorem is proved!
So, who is Fermat, what is the essence of the problem and has it really been solved? Pierre Fermat was born in 1601 in the family of a tanner, a wealthy and respected man - he served as second consul in his native town of Beaumont - this is something like an assistant to the mayor. Pierre studied first with the Franciscan monks, then at the Faculty of Law in Toulouse, where he then practiced advocacy. However, Fermat's range of interests went far beyond jurisprudence. He was especially interested in classical philology, his comments on the texts of ancient authors are known. And the second passion is mathematics.
In the 17th century, as, indeed, for many years later, there was no such profession: mathematician. Therefore, all the great mathematicians of that time were "part-time" mathematicians: Rene Descartes served in the army, Francois Viet was a lawyer, Francesco Cavalieri was a monk. There were no scientific journals then, and the classic of science Pierre Fermat did not publish a single scientific work during his lifetime. There was a rather narrow circle of "amateurs" who solved various interesting problems for them and wrote letters to each other about this, sometimes arguing (like Fermat with Descartes), but, basically, remained like-minded. They became the founders of new mathematics, the sowers of brilliant seeds, from which the mighty tree of modern mathematical knowledge began to grow, gaining strength and branching.
So, Fermat was the same "amateur". In Toulouse, where he lived for 34 years, everyone knew him, first of all, as an adviser to the Chamber of Investigation and an experienced lawyer. At the age of 30, he married, had three sons and two daughters, sometimes went on business trips, and during one of them he died suddenly at the age of 63. All! The life of this man, a contemporary of the Three Musketeers, is surprisingly uneventful and devoid of adventure. Adventures fell to the share of his Great Theorem. We will not talk about Fermat's entire mathematical heritage, and it is difficult to talk about him in a popular way. Take my word for it: this legacy is great and varied. The assertion that the Great Theorem is the pinnacle of his work is highly debatable. It's just that the fate of the Great Theorem is surprisingly interesting, and the vast world of people uninitiated in the mysteries of mathematics has always been interested not in the theorem itself, but in everything around it...
The roots of this whole story must be sought in antiquity, so beloved by Fermat. Approximately in the 3rd century, the Greek mathematician Diophantus lived in Alexandria, a scientist who thought in an original way, thinking outside the box and expressing his thoughts outside the box. Of the 13 volumes of his Arithmetic, only 6 have come down to us. Just when Fermat was 20 years old, a new translation of his works came out. Fermat was very fond of Diophantus, and these writings were his reference book. On its fields, Fermat wrote down his Great Theorem, which in its simplest modern form looks like this: the equation Xn + Yn = Zn has no solution in integers for n - more than 2. (For n = 2, the solution is obvious: Z2 + 42 = 52 ). In the same place, on the margins of the Diophantine volume, Fermat adds: "I discovered this truly wonderful proof, but these margins are too narrow for him."
At first glance, the little thing is simple, but when other mathematicians began to prove this "simple" theorem, no one succeeded for a hundred years. Finally, the great Leonhard Euler proved it for n = 4, then after 20 (!) years - for n = 3. And again the work stalled for many years. The next victory belongs to the German Peter Dirichlet (1805–1859) and the Frenchman Andrien Legendre (1752–1833), who admitted that Fermat was right for n = 5. Then the Frenchman Gabriel Lamet (1795–1870) did the same for n = 7. Finally, in the middle of the last century, the German Ernst Kummer (1810-1893) proved the Great Theorem for all values of n less than or equal to 100. Moreover, he proved it using methods that could not be known to Fermat, which further strengthened the veil of mystery around the Great Theorem.
Thus, it turned out that they were proving Fermat's theorem "piece by piece", but no one was able to "completely". New attempts at proofs only led to a quantitative increase in the values of n. Everyone understood that, having spent an abyss of labor, it was possible to prove the Great Theorem for an arbitrarily large number n, but Fermat spoke about any value of it greater than 2! It was in this difference between "arbitrarily large" and "any" that the whole meaning of the problem was concentrated.
However, it should be noted that attempts to prove Fermg's theorem were not just some kind of mathematical game, the solution of a complex rebus. In the course of these proofs, new mathematical horizons were opened up, problems arose and solved, which became new branches of the mathematical tree. The great German mathematician David Hilbert (1862-1943) cited the Great Theorem as an example of "what a stimulating effect a special and seemingly insignificant problem can have on science." The same Kummer, working on Fermat's theorem, himself proved theorems that formed the foundation of number theory, algebra and function theory. So proving the Great Theorem is not a sport, but a real science.
Time passed, and electronics came to the aid of professional "fsrmatnts". Electronic brains of new methods could not be invented, but they took speed. Around the beginning of the 80s, Fermat's theorem was proved with the help of a computer for n less than or equal to 5500. Gradually, this figure grew to 100,000, but everyone understood that such "accumulation" was a matter of pure technology, giving nothing to the mind or heart . They could not take the fortress of the Great Theorem "head on" and began to look for roundabout maneuvers.
In the mid-1980s, the young mathematician G. Filettings proved the so-called "Mordell's conjecture", which, by the way, was also "unreachable" by any of the mathematicians for 61 years. The hope arose that now, so to speak, "attacking from the flank", Fermat's theorem could also be solved. However, nothing happened then. In 1986, the German mathematician Gerhard Frei proposed a new proof method in Essesche. I do not undertake to explain it strictly, but not in mathematical, but in general human language, it sounds something like this: if we are convinced that the proof of some other theorem is an indirect, in some way transformed proof of Fermat's theorem, then, therefore, we will prove the Great Theorem. A year later, the American Kenneth Ribet from Berkeley showed that Frey was right and, indeed, one proof could be reduced to another. Many mathematicians around the world have taken this path. We have done a lot to prove the Great Theorem by Viktor Aleksandrovich Kolyvanov. The three-hundred-year-old walls of the impregnable fortress trembled. Mathematicians realized that it would not last long.
In the summer of 1993, in ancient Cambridge, at the Isaac Newton Institute of Mathematical Sciences, 75 of the world's most prominent mathematicians gathered to discuss their problems. Among them was the American professor Andrew Wiles of Princeton University, a prominent specialist in number theory. Everyone knew that he had been working on the Great Theorem for many years. Wiles made three presentations, and at the last one, on June 23, 1993, at the very end, turning away from the blackboard, he said with a smile:
I guess I won't continue...
There was dead silence at first, then a round of applause. Those sitting in the hall were qualified enough to understand: Fermat's Last Theorem is proved! In any case, none of those present found any errors in the above proof. Associate director of the Newton Institute, Peter Goddard, told reporters:
“Most experts didn't think they'd find out for the rest of their lives. This is one of the greatest achievements of mathematics of our century...
Several months have passed, no comments or denials followed. True, Wiles did not publish his proof, but only sent the so-called prints of his work to a very narrow circle of his colleagues, which, naturally, prevents mathematicians from commenting on this scientific sensation, and I understand Academician Ludwig Dmitrievich Faddeev, who said:
- I can say that the sensation happened when I see the proof with my own eyes.
Faddeev believes that the likelihood of Wiles winning is very high.
“My father, a well-known specialist in number theory, was, for example, sure that the theorem would be proved, but not by elementary means,” he added.
Another academician of ours, Viktor Pavlovich Maslov, was skeptical about the news, and he believes that the proof of the Great Theorem is not an actual mathematical problem at all. In terms of his scientific interests, Maslov, the chairman of the Council for Applied Mathematics, is far from "fermatists", and when he says that the complete solution of the Great Theorem is only of sporting interest, one can understand him. However, I dare to note that the concept of relevance in any science is a variable. 90 years ago, Rutherford, probably, was also told: "Well, well, well, the theory of radioactive decay ... So what? What is the use of it? .."
The work on the proof of the Great Theorem has already given a lot of mathematics, and one can hope that it will give more.
“What Wiles has done will move mathematicians into other areas,” said Peter Goddard. - Rather, this does not close one of the lines of thought, but raises new questions that will require an answer ...
Professor of Moscow State University Mikhail Ilyich Zelikin explained the current situation to me this way:
Nobody sees any mistakes in Wiles's work. But for this work to become a scientific fact, it is necessary that several reputable mathematicians independently repeat this proof and confirm its correctness. This is an indispensable condition for the recognition of Wiles' work by the mathematical community...
How long will it take for this?
I asked this question to one of our leading specialists in the field of number theory, Doctor of Physical and Mathematical Sciences Alexei Nikolaevich Parshin.
Andrew Wiles has a lot of time ahead of him...
The fact is that on September 13, 1907, the German mathematician P. Wolfskel, who, unlike the vast majority of mathematicians, was a rich man, bequeathed 100 thousand marks to the one who would prove the Great Theorem in the next 100 years. At the beginning of the century, interest from the bequeathed amount went to the treasury of the famous Getgangent University. This money was used to invite leading mathematicians to give lectures and conduct scientific work. At that time, David Hilbert, whom I have already mentioned, was chairman of the award commission. He did not want to pay the premium.
“Fortunately,” said the great mathematician, “it seems that we don’t have a mathematician, except for me, who would be able to do this task, but I will never dare to kill the goose that lays golden eggs for us.”
Before the deadline - 2007, designated by Wolfskel, there are few years left, and, it seems to me, a serious danger looms over "Hilbert's chicken". But it's not about the prize, actually. It's about the inquisitiveness of thought and human perseverance. They fought for more than three hundred years, but they still proved it!
And further. For me, the most interesting thing in this whole story is: how did Fermat himself prove his Great Theorem? After all, all today's mathematical tricks were unknown to him. And did he prove it at all? After all, there is a version that he seemed to have proved, but he himself found an error, and therefore he did not send the proofs to other mathematicians, but forgot to cross out the entry in the margins of the Diophantine volume. Therefore, it seems to me that the proof of the Great Theorem, obviously, took place, but the secret of Fermat's theorem remained, and it is unlikely that we will ever reveal it ...
Perhaps Fermat was mistaken then, but he was not mistaken when he wrote: “Perhaps posterity will be grateful to me for showing him that the ancients did not know everything, and this may penetrate the consciousness of those who will come after me. to pass the torch to his sons..."
