Ivliev Yu.A.
The article is devoted to the description of a fundamental mathematical error made in the process of proving Fermat's Last Theorem at the end of the 20th century. The detected error not only distorts the true meaning of the theorem, but also hinders the development of a new axiomatic approach to the study of powers of numbers and the natural series of numbers.
In 1995, an article was published that was similar in size to a book and reported on the proof of the famous Fermat's Great (Last) Theorem (WTF) (for the history of the theorem and attempts to prove it, see, for example,). After this event, there were many scientific articles and popular science books promoting this proof, however, in none of these works was a fundamental mathematical error revealed in it, which crept in not even through the fault of the author, but due to some strange optimism that seized the minds of mathematicians who dealt with this problem and related questions with her. The psychological aspects of this phenomenon have been investigated in. It also gives a detailed analysis of the oversight that occurred, which is not of a particular nature, but is the result of an incorrect understanding of the properties of the powers of integers. As shown in , Fermat's problem is rooted in a new axiomatic approach to the study of these properties, which has not yet been applied in modern science. But an erroneous proof stood in his way, giving number theorists false guidelines and leading researchers of Fermat's problem away from its direct and adequate solution. This work is devoted to removing this obstacle.
1. Anatomy of a mistake made during the proof of the WTF
In the process of very long and tedious reasoning, Fermat's original statement was reformulated in terms of a correspondence between a Diophantine equation of the p-th degree and elliptic curves of the 3rd order (see Theorems 0.4 and 0.5 in ). Such a comparison forced the authors of the de facto collective proof to announce that their method and reasoning lead to the final solution of Fermat's problem (recall that the WTF did not have recognized proofs for the case of arbitrary integer powers of integers until the 90s of the last century). The purpose of this consideration is to establish the mathematical incorrectness of the above comparison and, as a result of the analysis, to find a fundamental error in the proof presented in .
a) Where and what is wrong?
So, let's go through the text, where on p.448 it is said that after the "witty idea" of G. Frey (G. Frey), the possibility of proving the WTF has opened up. In 1984, G. Frey suggested and
K.Ribet later proved that the putative elliptic curve representing the hypothetical integer solution of Fermat's equation,
y 2 = x(x + u p)(x - v p) (1)
cannot be modular. However, A.Wiles and R.Taylor proved that any semistable elliptic curve defined over the field of rational numbers is modular. This led to the conclusion about the impossibility of integer solutions of Fermat's equation and, consequently, the validity of Fermat's statement, which in the notation of A. Wiles was written as Theorem 0.5: let there be an equality
u p+ v p+ w p = 0 (2)
where u, v, w- rational numbers, integer exponent p ≥ 3; then (2) is satisfied only if uvw = 0 .
Now, apparently, we should go back and critically consider why the curve (1) was a priori perceived as elliptic and what is its real relationship with Fermat's equation. Anticipating this question, A. Wiles refers to the work of Y. Hellegouarch, in which he found a way to associate Fermat's equation (presumably solved in integers) with a hypothetical 3rd order curve. Unlike G. Frey, I. Allegouches did not connect his curve with modular forms, but his method of obtaining equation (1) was used to further advance the proof of A. Wiles.
Let's take a closer look at work. The author conducts his reasoning in terms of projective geometry. Simplifying some of its notation and bringing them into line with , we find that the Abelian curve
Y 2 = X(X - β p)(X + γ p) (3)
the Diophantine equation is compared
x p+ y p+ z p = 0 (4)
where x, y, z are unknown integers, p is an integer exponent from (2), and the solutions of the Diophantine equation (4) α p , β p , γ p are used to write the Abelian curve (3).
Now, to make sure that this is a 3rd order elliptic curve, it is necessary to consider the variables X and Y in (3) on the Euclidean plane. To do this, we use the well-known rule of arithmetic of elliptic curves: if there are two rational points on a cubic algebraic curve and the line passing through these points intersects this curve at one more point, then the latter is also a rational point. Hypothetical equation (4) formally represents the law of addition of points on a straight line. If we make a change of variables x p = A, y p=B, z p = C and direct the straight line thus obtained along the X axis in (3), then it will intersect the 3rd degree curve at three points: (X = 0, Y = 0), (X = β p , Y = 0), (X = - γ p , Y = 0), which is reflected in the notation of the Abelian curve (3) and in a similar notation (1). However, is curve (3) or (1) really elliptical? Obviously not, because the segments of the Euclidean line, when adding points on it, are taken on a non-linear scale.
Returning to the linear coordinate systems of the Euclidean space, instead of (1) and (3) we obtain formulas that are very different from the formulas for elliptic curves. For example, (1) could be of the following form:
η 2p = ξ p (ξ p + u p)(ξ p - v p) (5)
where ξ p = x, η p = y, and the appeal to (1) in this case for the derivation of the WTF seems to be illegal. Despite the fact that (1) satisfies some criteria of the class of elliptic curves, it does not satisfy the most important criterion of being a 3rd degree equation in a linear coordinate system.
b) Error classification
So, once again we return to the beginning of the consideration and follow how the conclusion about the truth of the WTF is made. First, it is assumed that there is a solution of Fermat's equation in positive integers. Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (a plane curve of the 3rd degree) under the assumption that the elliptic curves thus obtained exist (the second unverified assumption). Thirdly, since it is proved by other methods that the constructed concrete curve is non-modular, it means that it does not exist. The conclusion follows from this: there is no integer solution of the Fermat equation and, therefore, the WTF is true.
There is one weak link in these arguments, which, after a detailed check, turns out to be a mistake. This mistake is made at the second stage of the proof process, when it is assumed that the hypothetical solution of Fermat's equation is also the solution of a third-degree algebraic equation describing an elliptic curve of a known form. In itself, such an assumption would be justified if the indicated curve were indeed elliptic. However, as can be seen from item 1a), this curve is presented in non-linear coordinates, which makes it “illusory”, i.e. not really existing in a linear topological space.
Now we need to clearly classify the found error. It lies in the fact that what needs to be proved is given as an argument of the proof. In classical logic, this error is known as the "vicious circle". In this case, the integer solution of the Fermat equation is compared (apparently, presumably uniquely) with a fictitious, non-existent elliptic curve, and then all the pathos of further reasoning goes to prove that a particular elliptic curve of this form, obtained from hypothetical solutions of the Fermat equation, does not exist.
How did it happen that such an elementary mistake was missed in a serious mathematical work? Probably, this happened due to the fact that “illusory” geometric figures of this type were not previously studied in mathematics. Indeed, who could be interested, for example, in a fictitious circle obtained from Fermat's equation by changing the variables x n/2 = A, y n/2 = B, z n/2 = C? After all, its equation C 2 = A 2 + B 2 has no integer solutions for integer x, y, z and n ≥ 3 . In non-linear coordinate axes X and Y, such a circle would be described by an equation that looks very similar to the standard form:
Y 2 \u003d - (X - A) (X + B),
where A and B are no longer variables, but concrete numbers determined by the above substitution. But if the numbers A and B are given their original form, which consists in their power character, then the heterogeneity of the notation in the factors on the right side of the equation immediately catches the eye. This sign helps to distinguish illusion from reality and to move from non-linear to linear coordinates. On the other hand, if we consider numbers as operators when comparing them with variables, as for example in (1), then both must be homogeneous quantities, i.e. must have the same degree.
Such an understanding of the powers of numbers as operators also makes it possible to see that the comparison of Fermat's equation with an illusory elliptic curve is not unambiguous. Take, for example, one of the factors on the right side of (5) and expand it into p linear factors by introducing a complex number r such that r p = 1 (see for example ):
ξ p + u p = (ξ + u)(ξ + r u)(ξ + r 2 u)...(ξ + r p-1 u) (6)
Then the form (5) can be represented as a decomposition into prime factors of complex numbers according to the type of algebraic identity (6), however, the uniqueness of such a decomposition in the general case is questionable, which was once shown by Kummer.
2. Conclusions
It follows from the previous analysis that the so-called arithmetic of elliptic curves is not capable of shedding light on where to look for the proof of the WTF. After the work, Fermat's statement, by the way, taken as the epigraph to this article, began to be perceived as a historical joke or practical joke. However, in reality it turns out that it was not Fermat who was joking, but experts who gathered at a mathematical symposium in Oberwolfach in Germany in 1984, at which G. Frey voiced his witty idea. The consequences of such a careless statement brought mathematics as a whole to the verge of losing its public confidence, which is described in detail in and which necessarily raises the question of the responsibility of scientific institutions to society before science. The mapping of the Fermat equation to the Frey curve (1) is the "lock" of Wiles's entire proof with respect to Fermat's theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof either.
Lately there have been various Internet reports that some prominent mathematicians have finally figured out Wiles' proof of Fermat's theorem, giving him an excuse in the form of a "minimal" recalculation of integer points in Euclidean space. However, no innovations can cancel the classical results already obtained by mankind in mathematics, in particular, the fact that although any ordinal number coincides with its quantitative counterpart, it cannot be a substitute for it in operations of comparing numbers with each other, and hence with inevitably follows the conclusion that the Frey curve (1) is not elliptic initially, i.e. is not by definition.
BIBLIOGRAPHY:
- Ivliev Yu.A. Reconstruction of the native proof of Fermat's Last Theorem - United Scientific Journal (section "Mathematics"). April 2006 No. 7 (167) p.3-9, see also Pratsi of the Luhansk branch of the International Academy of Informatization. Ministry of Education and Science of Ukraine. Shidnoukrainian National University named after. V. Dahl. 2006 No. 2 (13) pp.19-25.
- Ivliev Yu.A. The Greatest Science Scam of the 20th Century: The "Proof" of Fermat's Last Theorem Technical science(section "History and methodology of mathematics"). August 2007 No. 4 (30) pp. 34-48.
- Edwards G. (Edwards H.M.) Fermat's last theorem. Genetic introduction to algebraic number theory. Per. from English. ed. B.F. Skubenko. M.: Mir 1980, 484 p.
- Hellegouarch Y. Points d'ordre 2p h sur les courbes elliptiques - Acta Arithmetica. 1975 XXVI p.253-263.
- Wiles A. Modular elliptic curves and Fermat´s Last Theorem - Annals of Mathematics. May 1995 v.141 Second series No. 3 p.443-551.
Bibliographic link
Ivliev Yu.A. WILES' ERRONEOUS PROOF OF FERMAT'S GREAT THEOREM // Fundamental Research. - 2008. - No. 3. - P. 13-16;URL: http://fundamental-research.ru/ru/article/view?id=2763 (date of access: 03.03.2020). We bring to your attention the journals published by the publishing house "Academy of Natural History"
Grigory Perelman. Refusenik
Vasily Maksimov
In August 2006, the names of the best mathematicians on the planet 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, with brilliance he graduated from the famous 239th high school with an in-depth study of mathematics, he won numerous mathematical Olympiads: 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 Aleksandrov. 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 will 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 Poincaré 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 needed 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 this community, therefore he does not want to receive an award,” John Ball, the president of the World Union of Mathematicians (WCM), said at a press conference in Madrid.
Rumor has it 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.
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 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 notation of 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 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?
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 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, 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 "Principles 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 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 only remained to subordinate the logic created by Aristotle to algebra: then it would be possible, with the help of calculations, to prove the derivability or non-derivability of any scientific statements from a given set of axioms! 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 how to calculate the degree of ambiguity by methods of mathematical analysis, he discovered this dirty trick in the ring for degree 23. Gauss did not have time to learn about this version of exotic commutative algebra, but Gauss's students grew a new beautiful Theory of Ideals in place of another dirty trick. 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 certain 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 spanned a century and a half, creative biographies 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, the 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! You have to carefully write down the proof (but it will be many dozens of pages!) And put it aside 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 the reliable friends-colleagues and tell him the whole course of reasoning. From the outside, all the mistakes are more visible! Secondly, you need 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 with a VERBAL description 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 last hypothesis Ferma died at the age of three hundred and sixty, turning into a proven theorem, which will inevitably enter the textbooks of number theory.
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 the ancient era, 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. The spontaneous scientific and technological revolution needs a cultural shell to penetrate the minds of most people, and here comes science fiction first, 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 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"
For integers n greater than 2, the equation x n + y n = z n has no non-zero solutions in natural numbers.
You probably remember from your school days the Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. You may also remember the classic right triangle with sides whose lengths are related as 3: 4: 5. For it, the Pythagorean theorem looks like this:
This is an example of solving the generalized Pythagorean equation in non-zero integers for n = 2. The Great Theorem Fermat (also called "Fermat's Last Theorem" and "Fermat's Last Theorem") is the assertion that, for values n> 2 equations of the form x n + y n = z n do not have nonzero solutions in natural numbers.
The history of Fermat's Last Theorem is very entertaining and instructive, and not only for mathematicians. Pierre de Fermat contributed to the development of various areas of mathematics, but the main part of his scientific heritage was published only posthumously. The fact is that mathematics for Fermat was something like a hobby, not a professional occupation. He corresponded with the leading mathematicians of his time, but did not seek to publish his work. Scientific works The farm is mostly found in the form of private correspondence and fragmentary notes, often made in the margins of various books. It is on the margins (of the second volume of the ancient Greek Arithmetic by Diophantus. - Note. translator) shortly after the death of the mathematician, the descendants discovered the formulation of the famous theorem and the postscript:
« I found a truly wonderful proof of this, but these margins are too narrow for him.».
Alas, apparently, Fermat did not bother to write down the “miraculous proof” he found, and the descendants unsuccessfully searched for him for three years. superfluous century. Of all Fermat's disparate scientific heritage, containing many surprising statements, it was the Great Theorem that stubbornly resisted solution.
Whoever did not take up the proof of Fermat's Last Theorem - all in vain! Another great French mathematician, René Descartes (René Descartes, 1596-1650), called Fermat a "braggart", and the English mathematician John Wallis (John Wallis, 1616-1703) called him a "damn Frenchman". Fermat himself, however, nevertheless left behind a proof of his theorem for the case n= 4. With proof for n= 3 was solved by the great Swiss-Russian mathematician of the 18th century Leonard Euler (1707–83), after which, having failed to find proofs for n> 4, jokingly offered to search Fermat's house to find the key to the lost evidence. In the 19th century, new methods of number theory made it possible to prove the statement for many integers within 200, but, again, not for all.
In 1908 a prize of DM 100,000 was established for this task. The prize fund was bequeathed to the German industrialist Paul Wolfskehl, who, according to legend, was about to commit suicide, but was so carried away by Fermat's Last Theorem that he changed his mind about dying. With the advent of adding machines, and then computers, the bar of values n began to rise higher and higher - up to 617 by the beginning of World War II, up to 4001 in 1954, up to 125,000 in 1976. At the end of the 20th century, the most powerful computers of military laboratories in Los Alamos (New Mexico, USA) were programmed to solve the Fermat problem in the background (similar to the screen saver mode of a personal computer). Thus, it was possible to show that the theorem is true for incredibly large values x, y, z and n, but this could not serve as a rigorous proof, since any of the following values n or triplets natural numbers could disprove the theorem as a whole.
Finally, in 1994, the English mathematician Andrew John Wiles (Andrew John Wiles, b. 1953), while working at Princeton, published a proof of Fermat's Last Theorem, which, after some modifications, was considered exhaustive. The proof took more than a hundred magazine pages and was based on the use of the modern apparatus of higher mathematics, which had not been developed in Fermat's era. So what, then, did Fermat mean by leaving a message in the margins of the book that he had found proof? Most of the mathematicians I have spoken to on this subject have pointed out that over the centuries there have been more than enough incorrect proofs of Fermat's Last Theorem, and that it is likely that Fermat himself found a similar proof but failed to see the error in it. However, it is possible that there is still some short and elegant proof of Fermat's Last Theorem, which no one has yet found. Only one thing can be said with certainty: today we know for sure that the theorem is true. Most mathematicians, I think, would unreservedly agree with Andrew Wiles, who remarked about his proof, "Now at last my mind is at peace."
That the 2016 Abel Prize will go to Andrew Wiles for his proof of the Taniyama-Shimura conjecture for semistable elliptic curves and the proof of Fermat's Last Theorem that follows from this conjecture. Currently, the premium is 6 million Norwegian kroner, that is, approximately 50 million rubles. According to Wiles, the award came as a "complete surprise" to him.
Fermat's theorem, proved more than 20 years ago, still attracts the attention of mathematicians. In part, this is due to its formulation, which is understandable even to a schoolboy: to prove that for natural numbers n>2 there are no such triples of non-zero integers that a n + b n = c n . Pierre de Fermat wrote this expression in the margins of Diophantus' Arithmetic, with the remarkable caption "I have found a truly wonderful proof [of this assertion] for this, but the margins of the book are too narrow for it." Unlike most math tales, this one is real.
The presentation of the award is a great occasion to recall ten entertaining stories related to Fermat's theorem.
1.
Before Andrew Wiles proved Fermat's theorem, it was more properly called a conjecture, that is, Fermat's hypothesis. The fact is that a theorem is, by definition, an already proven statement. However, for some reason, just such a name stuck to this statement.
2.
If we put n = 2 in Fermat's theorem, then such an equation has infinitely many solutions. These solutions are called "Pythagorean triples". They got this name because they correspond to right-angled triangles, the sides of which are expressed by just such sets of numbers. You can generate Pythagorean triples using these three formulas (m 2 - n 2, 2mn, m 2 + n 2). It is necessary to substitute different values of m and n into these formulas, and as a result we will get the triples we need. The main thing here, however, is to make sure that the resulting numbers will be greater than zero - lengths cannot be expressed as negative numbers.
By the way, it is easy to see that if all numbers in a Pythagorean triple are multiplied by some non-zero number, a new Pythagorean triple will be obtained. Therefore, it is reasonable to study triples in which the three numbers in the aggregate do not have a common divisor. The scheme that we have described makes it possible to obtain all such triples - this is by no means a simple result.
3.
On March 1, 1847, at a meeting of the Paris Academy of Sciences, two mathematicians at once - Gabriel Lame and Augustin Cauchy - announced that they were on the verge of proving a remarkable theorem. They ran a race to publish pieces of evidence. Most academics cheered for Lame, because Cauchy was a self-righteous, intolerant religious fanatic (and, of course, an absolutely brilliant part-time mathematician). However, the match was not destined to end - through his friend Joseph Liouville, the German mathematician Ernst Kummer informed the academicians that there was one and the same error in the proofs of Cauchy and Lame.
At school, it is proved that the decomposition of a number into prime factors is unique. Both mathematicians believed that if you look at the decomposition of integers already in the complex case, then this property - uniqueness - will be preserved. However, it is not.
It is noteworthy that if we consider only m + i n, then the decomposition is unique. Such numbers are called Gaussian. But Lame and Cauchy's work required factoring in cyclotomic fields. These are, for example, numbers in which m and n are rational, and i satisfies the property i^k = 1.
4.
Fermat's theorem for n = 3 has a clear geometric meaning. Let's imagine that we have many small cubes. Suppose we have collected two large cubes from them. In this case, of course, the sides will be integers. Is it possible to find two such large cubes that, having disassembled them into their component small cubes, we could assemble one large cube from them? Fermat's Theorem says that this can never be done. It's funny that if you ask the same question for three cubes, the answer is yes. For example, there is such a quadruple of numbers, discovered by the wonderful mathematician Srinivas Ramanujan:
3 3 + 4 3 + 5 3 = 6 3
5.
Leonhard Euler was noted in the history of Fermat's theorem. He did not really succeed in proving the statement (or even approaching the proof), but he formulated the hypothesis that the equation
x 4 + y 4 + z 4 = u 4
has no solution in integers. All attempts to find a direct solution to such an equation turned out to be fruitless. It wasn't until 1988 that Nahum Elkies of Harvard managed to find a counterexample. It looks like this:
2 682 440 4 + 15 365 639 4 + 18 796 760 4 = 20 615 673 4 .
Usually this formula is remembered in the context of a numerical experiment. As a rule, in mathematics it looks like this: there is some formula. The mathematician checks this formula in simple cases, convinces himself of the truth and formulates some hypothesis. Then he (although more often some of his graduate students or students) writes a program in order to check that the formula is correct for sufficiently large numbers that cannot be counted by hand (we are talking about one such experiment with prime numbers). This is not a proof, of course, but an excellent reason to declare a hypothesis. All these constructions are based on the reasonable assumption that if there is a counterexample to some reasonable formula, then we will find it quickly enough.
Euler's conjecture reminds us that life is much more diverse than our fantasies: the first counterexample can be arbitrarily large.
6.
In fact, of course, Andrew Wiles wasn't trying to prove Fermat's Theorem - he was solving a more difficult problem called the Taniyama-Shimura conjecture. There are two remarkable classes of objects in mathematics. The first one is called modular forms and is essentially a function on the Lobachevsky space. These functions do not change during the movements of this very plane. The second is called "elliptic curves" and is the curves given by the equation of the third degree in the complex plane. Both objects are very popular in number theory.
In the 1950s, two talented mathematicians Yutaka Taniyama and Goro Shimura met in the library of the University of Tokyo. At that time, there was no special mathematics at the university: it simply did not have time to recover after the war. As a result, scientists studied using old textbooks and discussed at seminars problems that in Europe and the USA were considered solved and not particularly relevant. It was Taniyama and Shimura who discovered that there is a correspondence between modular forms and elliptic functions.
They tested their conjecture on some simple classes of curves. It turned out that it works. So they suggested that this connection is always there. This is how the Taniyama-Shimura hypothesis appeared, and three years later Taniyama committed suicide. In 1984, the German mathematician Gerhard Frey showed that if Fermat's Theorem is wrong, then the Taniyama-Shimura conjecture is wrong. It followed from this that the one who proved this conjecture would also prove the theorem. This is exactly what he did - though not quite in general view- Wiles.
7.
Wiles spent eight years proving the conjecture. And during the check, the reviewers found an error in it, which “killed” most of the proof, nullifying all the years of work. One of the reviewers, by the name of Richard Taylor, undertook to repair the hole with Wiles. While they were working, a message appeared that Elkies, the same one who found a counterexample to Euler's conjecture, also found a counterexample to Fermat's theorem (later it turned out that this was an April Fool's joke). Wiles fell into a depression and did not want to continue - the hole in the evidence could not be closed in any way. Taylor talked Wiles into wrestling for another month.
A miracle happened and by the end of the summer mathematicians managed to make a breakthrough - this is how the works "Modular elliptic curves and Fermat's Last Theorem" by Andrew Wiles (pdf) and "Ring-theoretic properties of some Hecke algebras" by Richard Taylor and Andrew Wiles were born. This was the correct proof. It was published in 1995.
8.
In 1908, the mathematician Paul Wolfskel died in Darmstadt. After himself, he left a will in which he gave the mathematical community 99 years to find a proof of Fermat's Last Theorem. The author of the proof should have received 100 thousand marks (by the way, the author of the counterexample would not have received anything). According to a popular legend, it was love that prompted the Wolfskell mathematicians to make such a gift. Here is how Simon Singh describes the legend in his book Fermat's Last Theorem:
The story begins with Wolfskel getting carried away beautiful woman, whose identity has never been established. Much to Wolfskel's regret, the mysterious woman rejected him. He fell into such deep despair that he decided to commit suicide. Wolfskel was a passionate man, but not impulsive, and therefore began to work out his death in every detail. He set a date for his suicide and decided to shoot himself in the head with the first strike of the clock at exactly midnight. During the remaining days, Wolfskel decided to put his affairs in order, which were going great, and on the last day he made a will and wrote letters to close friends and relatives.
Wolfskehl worked so diligently that he finished all his business before midnight and, in order to somehow fill the remaining hours, he went to the library, where he began to look through mathematical journals. He soon came across Kummer's classic paper explaining why Cauchy and Lame had failed. Kummer's work was one of the most significant mathematical publications of his century and was the best reading for a mathematician who was contemplating suicide. Wolfskel carefully, line by line, followed Kummer's calculations. Unexpectedly, it seemed to Wolfskel that he had discovered a gap: the author made a certain assumption and did not substantiate this step in his reasoning. Wolfskehl wondered if he had really found a serious gap, or if Kummer's assumption was justified. If a gap was found, then there was a chance that Fermat's Last Theorem could be proved much easier than many thought.
Wolfskehl sat down at the table, carefully analyzed the “flawed” part of Kummer’s reasoning and began to sketch out a mini-proof, which was supposed to either support Kummer’s work or demonstrate the fallacy of the assumption he made and, as a result, refute all his arguments. By dawn, Wolfskehl had finished his calculations. The bad (mathematically) news was that Kummer's proof had been healed, and Fermat's Last Theorem was still out of reach. But there was good news: the time for suicide had passed, and Wolfskehl was so proud that he had managed to find and fill a gap in the work of the great Ernest Kummer that his despair and sadness dispelled themselves. Mathematics gave him back the thirst for life.
However, there is an alternative version. According to her, Wolfskel took up mathematics (and, in fact, Fermat's theorem) because of progressive multiple sclerosis, which prevented him from doing what he loved - being a doctor. And he left money to mathematicians so as not to leave his wife, whom he simply hated by the end of his life.
9.
Attempts to prove Fermat's theorem by elementary methods led to the emergence of a whole class of strange people called "fermatists". They were engaged in the fact that they produced a huge amount of evidence and did not despair at all when they found an error in these proofs.
At the Mechanics and Mathematics Department of Moscow State University legendary character by the name of Dobretsov. He collected certificates from various departments and, using them, penetrated the mekhmat. This was done solely in order to find the victim. Somehow he came across a young graduate student (the future academician Novikov). He, in his naivete, began to carefully study the stack of papers that Dobretsov slipped him with the words, they say, here is the proof. After another "here's a mistake ..." Dobretsov took the stack and stuffed it into his briefcase. From the second briefcase (yes, he walked around the mekhmat with two briefcases), he took out the second stack, sighed and said: "Well, then let's see option 7 B."
By the way, most of these proofs begin with the phrase "Let's move one of the terms to the right side of the equality and factorize it."
10.
The story about the theorem would be incomplete without the wonderful film "The Mathematician and the Devil".
Amendment
Section 7 of this paper originally stated that Nahum Elkies had found a counterexample to Fermat's Theorem, which later turned out to be wrong. This is not true: the message about the counterexample was an April Fool's joke. We apologize for the inaccuracy.
Andrey Konyaev