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Who solved the farm equation? The proof of Fermat's theorem is elementary, simple, and understandable. Work by Shimura and Taniyama

Who solved the farm equation? The proof of Fermat's theorem is elementary, simple, and understandable. Work by Shimura and Taniyama

For integers n greater than 2, the equation x n + y n = z n has no nonzero solutions in natural numbers.

You probably remember from your school days Pythagorean theorem: square of the hypotenuse of a right triangle equal to the sum squares of legs. You may also remember the classic right triangle with sides whose lengths are in the ratio 3: 4: 5. For it, the Pythagorean theorem looks like this:

This is an example of solving the generalized Pythagorean equation in nonzero integers with n = 2. Great Theorem Fermat (also called Fermat's Last Theorem and Fermat's Last Theorem) states that for the values n> 2 equations of the form x n + y n = z n have no non-zero solutions in natural numbers.

The history of Fermat's Last Theorem is very interesting and instructive, and not only for mathematicians. Pierre de Fermat contributed to the development of various fields of mathematics, but the main part of his scientific legacy was published only posthumously. The fact is that mathematics for Fermat was something of a hobby, and not a professional occupation. He corresponded with the leading mathematicians of his time, but did not strive to publish his work. Scientific works The farm is mainly found in the form of private correspondence and fragmentary notes, often written in the margins of various books. It is in the margins (of the second volume of the ancient Greek “Arithmetic” of Diophantus. - Note translator) soon 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 fields are too narrow for it».

Alas, apparently, Fermat never bothered to write down the “miraculous proof” he found, and descendants unsuccessfully searched for it for more than three centuries. Of all Fermat's scattered scientific heritage, which contains many surprising statements, it was the Great Theorem that stubbornly refused to be solved.

Whoever has tried to prove Fermat's Last Theorem is in vain! Another great French mathematician, René Descartes (1596–1650), called Fermat a “braggart,” and the English mathematician John Wallis (1616–1703) called him a “damn Frenchman.” Fermat himself, however, still 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 Leonhard Euler (1707–83), after which, unable to find evidence for n> 4, jokingly suggested that Fermat's house be searched to find the key to the lost evidence. In the 19th century, new methods in number theory made it possible to prove the statement for many integers within 200, but again, not for all.

In 1908, a prize of 100,000 German marks was established for solving this problem. The prize fund was bequeathed by the German industrialist Paul Wolfskehl, who, according to legend, was going 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 value bar n began to rise higher and higher - to 617 by the beginning of World War II, to 4001 in 1954, to 125,000 in 1976. At the end of the 20th century, the most powerful computers at military laboratories in Los Alamos (New Mexico, USA) were programmed to solve Fermat's 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 strict proof, since any of the following values n or threes natural numbers could refute the theorem as a whole.

Finally, in 1994, the English mathematician Andrew John Wiles (b. 1953), working at Princeton, published a proof of Fermat's Last Theorem, which, after some modifications, was considered comprehensive. The proof took more than a hundred journal pages and was based on the use of modern apparatus higher mathematics, which was not 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 the proof? Most of the mathematicians with whom I spoke on this topic pointed out that over the centuries there had been more than enough incorrect proofs of Fermat's Last Theorem, and that, most likely, Fermat himself had found a similar proof, but failed to recognize the error in it. However, it is possible that there is still some short and elegant proof of Fermat’s Last Theorem that 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 agree unreservedly with Andrew Wiles, who remarked of his proof: “Now at last my mind is at peace.”

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 twentieth century. The discovered 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, similar in size to a book, and reporting 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, many scientific articles and popular science books promoting this proof, but not one of these works revealed a fundamental mathematical error in it, which crept in not even through the fault of the author, but due to some strange optimism that gripped the minds of mathematicians who dealt with this problem and related ones with her questions. Psychological aspects This phenomenon has been studied in. Here we provide a detailed analysis of the mistake that occurred, which is not of a private nature, but is a consequence of an incorrect understanding of the properties of powers of integers. As shown in, Fermat's problem is rooted in a new axiomatic approach to the study of these properties, which is still in modern science was not used. But an erroneous proof stood in his way, providing number theory specialists with false guidelines and leading researchers of Fermat’s problem away from its direct and adequate solution. this work is dedicated to eliminating this obstacle.

1. Anatomy of an error made during the WTF proof

In the process of very long and tedious reasoning, Fermat's original statement was reformulated in terms of a comparison of a Diophantine equation of the pth degree with elliptic curves of the 3rd order (see Theorems 0.4 and 0.5 in). This comparison forced the authors of the virtually collective proof to announce that their method and reasoning lead to a final solution to 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 the error?

So, we will follow the text, where on p. 448 it is said that after the “witty idea” of G. Frey, the possibility of proving the WTF opened up. In 1984, G. Frey suggested and

K. Ribet later proved that the supposed 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 every 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, about 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, whole indicator p ≥ 3; then (2) is satisfied only if uvw = 0 .

Now, apparently, we should go back and critically think about why curve (1) was a priori perceived as elliptic and what is its real connection 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 third-order curve. Unlike G. Frey, I. Elleguarche did not connect his curve with modular forms, however, 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 notations 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 the 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 an elliptic curve of the 3rd order, it is necessary to consider the variables X and Y in (3) in 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 a line passing through these points intersects this curve at another point, then the latter is also a rational point. Hypothetical equation (4) formally represents the law of adding points on a straight line. If we make a change of variables x p = A, y p = B, z p = C and direct the resulting straight line 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) actually elliptic? Obviously, no, because the segments of the Euclidean line, when adding points on it, are taken on a nonlinear scale.

Returning to the linear coordinate systems of Euclidean space, we obtain instead of (1) and (3) 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 to derive the WTF seems illegitimate. Despite the fact that (1) satisfies some criteria for the class of elliptic curves, the most important criterion is to be an equation of the 3rd degree in linear system it does not satisfy the coordinates.

b) Error classification

So, let us once again return to the beginning of the consideration and see how the conclusion about the truth of the WTF is reached. First, it is assumed that there is some solution to Fermat's equation in positive integers. Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (a plane curve of degree 3) under the assumption that the elliptic curves thus obtained exist (the second unconfirmed assumption). Thirdly, since other methods prove that the particular curve constructed is non-modular, it means that it does not exist. This leads to the conclusion: there is no integer solution to Fermat’s equation and, therefore, the WTF is correct.

There is one weak link in these arguments, which, after detailed verification, turns out to be an error. This error is made at the second stage of the proof process, when it is assumed that the hypothetical solution to Fermat's equation is also the solution to an algebraic equation of the 3rd degree describing an elliptic curve of a known form. In itself, such an assumption would be justified if the indicated curve were really elliptic. However, as can be seen from point 1a), this curve is presented in nonlinear coordinates, which makes it “illusory”, i.e. not really existing in linear topological space.

Now we need to clearly classify the found error. It lies in the fact that what needs to be proven is presented as an argument of proof. In classical logic this error is known as a "vicious circle". IN in this case the integer solution of Fermat's equation is compared (apparently, presumably uniquely) with a fictitious, non-existent elliptic curve, and then all the pathos of further reasoning is spent on proving that a specific elliptic curve of this kind, obtained from hypothetical solutions of Fermat's equation, does not exist.

How did it happen that such an elementary error was missed in serious mathematical work? This probably happened due to the fact that “illusory” objects had not previously been studied in mathematics. geometric figures of the specified type. Indeed, who could be interested, for example, in a fictitious circle obtained from Fermat’s equation by replacing the variables x n/2 = A, y n/2 = B, z n/2 = C? After all, its equation C 2 = A 2 + B 2 does not have integer solutions for integer x, y, z and n ≥ 3. In the nonlinear coordinate axes X and Y, such a circle would be described by the equation, according to appearance very similar to the standard form:

Y 2 = - (X - A)(X + B),

where A and B are no longer variables, but specific 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 notation in the factors on the right side of the equation immediately catches the eye. This feature helps to distinguish illusion from reality and move from nonlinear 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 degrees.

This understanding of powers of numbers as operators also allows us to see that the comparison of the Fermat equation with an illusory elliptic curve is not unambiguous. Take, for example, one of the factors on the right side of (5) and decompose it into p linear factors, 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 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 in question, as was once shown by Kummer.

2. Conclusions

From the previous analysis it follows that the so-called arithmetic of elliptic curves is not able to shed light on where to look for a 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 historical joke or a joke. However, in reality it turns out that it was not Fermat who joked, but the specialists 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 brink of losing its public trust, which is described in detail in and which necessarily raises the question of the responsibility of scientific institutions to society. The comparison of the Fermat equation with the Frey curve (1) is the “lock” of Wiles’ entire proof regarding Fermat’s theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof.

IN Lately There are various Internet reports that some prominent mathematicians have finally figured out Wiles's proof of Fermat's theorem, having come up with a justification for it 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 analogue, it cannot be a replacement for it in operations of comparing numbers with each other, and hence with the inevitable conclusion follows that the Frey curve (1) is not initially elliptic, i.e. is not it 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) pp. 3-9, see also Praci Lugansk Branch of the International Academy of Informatization. Ministry of Education and Science of Ukraine. Skhidnoukransky National University named after. V.Dal. 2006 No. 2 (13) p.19-25.
Ivliev Yu.A. The greatest scientific scam of the 20th century: the “proof” of Fermat’s Last Theorem - Natural and engineering sciences (section “History and methodology of mathematics”). August 2007 No. 4 (30) p.34-48.
Edwards G. (Edwards H.M.) Fermat's last theorem. Genetic introduction to algebraic number theory. Per. from English edited by 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. WILLES' FALSE PROOF OF FERMA'S LAST THEOREM // Basic Research. – 2008. – No. 3. – P. 13-16;
URL: http://fundamental-research.ru/ru/article/view?id=2763 (access date: 03/17/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"

Many years ago I received a letter from Tashkent from Valery Muratov, judging by the handwriting, a man of adolescence, who then lived on Kommunisticheskaya Street at number 31. The guy was determined: “Get straight to the point. How much will you pay me for proving Fermat’s theorem? suits at least 500 rubles. At another time, I would prove 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. More less people can even describe his great theorem in the most general terms. But everyone knows that there is some kind of Fermat’s theorem, the proof of which mathematicians around the world have been struggling for more than 300 years, but cannot prove!

There are many ambitious people, and the very consciousness that there is something that others cannot do spurs their ambition even more. Therefore, at the academy, scientific institutes and even the editors of newspapers all over the world have come and are coming thousands (!) of proofs of the Great Theorem - an unprecedented and never broken record of pseudoscientific amateur activity. There is even a term: “Fermatists,” that is, people obsessed with proving the Great Theorem, who completely tormented 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 wrote down the page number. And then in the summer of 1994, newspapers around the world reported something completely sensational: the Great Theorem had been proven!

So, who is Fermat, what is the problem, and is it really solved? Pierre Fermat was born in 1601 into the family of a tanner, a wealthy and respected man - he served as second consul in his hometown of Beaumont - 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 law. However, Fermat's range of interests went far beyond jurisprudence. He was especially interested in classical philology, and his commentaries on the texts of ancient authors are known. And my 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 mathematicians “part-time”: Rene Descartes served in the army, François Viète was a lawyer, Francesco Cavalieri was a monk. Scientific journals there was none then, and the classic scientist Pierre Fermat did not publish a single scientific work during his lifetime. There was a fairly narrow circle of “amateurs” who solved various problems that were interesting to them and wrote letters to each other about this, sometimes argued (like Fermat and Descartes), but mostly remained like-minded. They became the founders of new mathematics, 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 investigative chamber 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. The adventures came with his Great Theorem. Let’s not talk about Fermat’s entire mathematical heritage, and it’s difficult to talk about it popularly. Take my word for it: this heritage is great and varied. The claim that the Great Theorem is the pinnacle of his work is highly controversial. 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 have 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. Around the 3rd century, the Greek mathematician Diophantus lived in Alexandria, an original scientist who thought outside the box and expressed his thoughts outside the box. Of the 13 volumes of his Arithmetic, only 6 have reached us. Just when Fermat turned 20, new translation his writings. Fermat was very interested in Diophantus, and these works were his reference book. It was in its fields that Fermat wrote down his Great Theorem, which in its simplest form modern form looks like this: the equation Xn + Yn = Zn has no solution in integers for n - greater than 2. (For n = 2, the solution is obvious: 32 + 42 = 52). There, in the margins of the Diophantine volume, Fermat adds: “I have discovered this truly wonderful proof, but these margins are too narrow for it.”

At first glance, this is a simple thing, 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 20 (!) years later - for n = 3. And again the work stalled for many years. The next victory belonged to the German Peter Dirichlet (1805-1859) and the Frenchman Andrien Legendre (1752-1833) - they admitted that Fermat was right for n = 5. Then the Frenchman Gabriel Lamé (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 Fermat could not have known, which further increased the flair of mystery around the Great Theorem.

Thus, it turned out that they proved Fermat’s theorem “piece by piece,” but no one succeeded “in its entirety.” New attempts at proofs only led to a quantitative increase in the values of n. Everyone understood that, with a lot of work, it was possible to prove the Great Theorem for arbitrarily large number n, but Fermat spoke about any value greater than 2! It was in this difference between “as much as you like” 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, solving a complex rebus. In the process of these proofs, new mathematical horizons were opened, problems arose and were solved, becoming new branches of the mathematical tree. The great German mathematician David Hilbert (1862-1943) cited the Great Theorem as an example of “the stimulating influence on science that a special and seemingly insignificant problem can have.” 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 “fsrmatntsts”. Electronic brains couldn’t come up with new methods, but they did it quickly. Around the beginning of the 80s, Fermat’s theorem was proven 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 workaround maneuvers.

In the mid-80s, a young non-mathematician G. Filytings proved the so-called “Mordell conjecture”, which, by the way, also “didn’t come into the hands” of any mathematician for 61 years. The hope arose that now, by “attacking from the flank,” so to speak, Fermat’s theorem could be solved. However, nothing happened then. In 1986, the German mathematician Gerhard Frey proposed a new proof method in Essence. I don’t undertake to explain it strictly, but not in a mathematical, but in a universal 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 can be reduced to another. Many mathematicians followed this path. different countries peace. Viktor Aleksandrovich Kolyvanov has done a lot to prove the Great Theorem. The three-hundred-year-old walls of the impregnable fortress began to shake. Mathematicians realized that it would not stand for long.

In the summer of 1993 in ancient Cambridge, at the Institute mathematical sciences named after Isaac Newton, 75 of the world's most prominent mathematicians gathered to discuss their problems. Among them was the American professor Andrew Wiles from Princeton University, a major specialist in number theory. Everyone knew that he had been studying the Great Theorem for many years. Wiles gave three reports and at the last one - June 23, 1993 - at the very end, turning away from the board, he said with a smile:

- I guess I won’t continue...

At first there was dead silence, then a deluge of applause. Those sitting in the hall were qualified enough to understand: Fermat's Last Theorem was proven! In any case, none of those present found any errors in the evidence presented. Deputy Director of the Newton Institute Peter Goddard told reporters:

“Most experts didn’t think they would know the answer until the end of their lives.” This is one of the greatest achievements in mathematics of our century...

Several months passed, no comments or refutations were made. True, Wiles did not publish his proof, but only sent out 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 a sensation has occurred 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, confident that the theorem would be proven, but not by elementary means,” he added.

Our other academician, Viktor Pavlovich Maslov, was skeptical about the news, and believes that the proof of the Great Theorem is not a pressing mathematical problem at all. In terms of his scientific interests, Maslov, the chairman of the council on applied mathematics, is far from the “fermatists,” and when he says that complete solution The Great Theorem is only of sporting interest, it can be understood. However, I dare to note that the concept of relevance in any science is a variable quantity. 90 years ago, Rutherford was probably also told: “Well, okay, well, the theory of radioactive decay... So what? What’s the use of it?..”

The work on the proof of the Great Theorem has already given a lot to mathematics, and we can hope that it will give more.

“What Wiles did will advance mathematicians into other fields,” said Peter Goddard. — Rather, it does not close one of the directions of thought, but raises new questions that will require an answer...

Moscow State University professor Mikhail Ilyich Zelikin explained the current situation to me this way:

Nobody sees any mistakes in Wiles' work. But for this work to become scientific fact, it is necessary for several reputable mathematicians to independently repeat this proof and confirm its correctness. This is an indispensable condition for the mathematical public to understand Wiles' work...

How long will it take?

I asked this question to one of our leading experts in the field of number theory, Doctor of Physical and Mathematical Sciences Alexey Nikolaevich Parshin.

— Andrew Wiles still has a lot of time ahead...

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 on the bequeathed amount went to the treasury of the famous University of Goethanghent. With this money, leading mathematicians were invited to give lectures, scientific work. At that time, the chairman of the award committee was the already mentioned David Gilbert. He really didn’t want to pay the bonus.

“Fortunately,” he said great mathematician, - it seems that we do not have a mathematician, except me, who would be able to do this task, but I will never dare to kill the goose that lays golden eggs for us -

There are few years left until the deadline of 2007, designated by Wolfskehl, and, it seems to me, a serious danger looms over “Hilbert’s chicken”. But it’s not really about the bonus. It's a matter of 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 it, but he himself found an error, and therefore did not send the proof to other mathematicians, and forgot to cross out the entry in the margins of Diophantus’s volume. Therefore, it seems to me that the proof of the Great Theorem has obviously taken place, but the secret of Fermat’s theorem remains, and it is unlikely that we will ever reveal it...

Fermat may have been mistaken then, but he was not mistaken when he wrote: “Perhaps posterity will be grateful to me for showing them that the ancients did not know everything, and this may penetrate the consciousness of those who come after me to pass the torch to his sons..."

There are not many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only math problem, which became so widely known and became a real legend. It is mentioned in many books and films, and the main context of almost all mentions is the impossibility of proving the theorem.

Yes, this theorem is very well known 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 called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in essence and understandable to anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.

Why is she so famous? Now we'll find out...

Are there many proven, unproven and as yet unproven theorems? The point here is that Fermat's Last Theorem represents the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by anyone with a 5th grade level. high school, but the proof is not even for every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in mathematics, is there a single problem that could be formulated so simply, but remained unsolved for so long. 2. What does it consist of?

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

In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triplets satisfying the equality x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas to find them. They probably tried to look for threes or more high degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the brotherhood were more philosophers and aesthetes than mathematicians.

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

Starting from 3, 4, 5 - indeed, a junior student understands that 9 + 16 = 25.

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

So, it turns out that they are NOT. This is where the trick begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, its absence. When you need to prove that there is a solution, you can and should simply present this solution.

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

Say: “I haven’t found such solutions”? Or maybe you weren't looking well? What if they exist, only very large, very large, such that even a super-powerful computer still doesn’t have enough strength? This is what is difficult.

This can be shown visually like this: if you take two squares of suitable sizes and disassemble them into unit squares, then from this bunch of unit squares you get a third square (Fig. 2):

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

But the 17th century mathematician Frenchman Pierre de Fermat enthusiastically explored general equation x n +y n =z n . And finally, I concluded: for n>2 there are no integer solutions. Fermat's proof is irretrievably lost. Manuscripts are burning! All that remains is his remark in Diophantus’ Arithmetic: “I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it.”

Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never making mistakes. Even if he did not leave evidence of a statement, it was subsequently confirmed. Moreover, Fermat proved his thesis for n=4. Thus, the hypothesis of the French mathematician went down in history as Fermat’s Last Theorem.

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

Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the proof for n = 7) and many others. By the mid-1980s it became clear that scientific world is on the way to the final solution of Fermat's Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat's last theorem was practically over.

It is easily shown that it is enough to prove Fermat’s theorem only for simple n: 3, 5, 7, 11, 13, 17, ... For composite n, the proof remains valid. But also prime numbers infinitely many...

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

Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that, using the methods of mathematics of the 19th century, the theorem in general view cannot be proven. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unawarded.

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

He soon died of natural causes. The heirs were quite surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal scientific society Göttingen, which in the same year announced a competition for the Wolfskehl Prize. 100,000 marks were awarded to the person who proved Fermat's theorem. Not a pfennig was awarded for refuting the theorem...

Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and resolutely refused to waste time on such a useless exercise. But the amateurs had a blast. A few weeks after the announcement, an avalanche of “evidence” hit the University of Göttingen. Professor E.M. Landau, whose responsibility was to analyze the evidence sent, distributed cards to his students:

Dear. . . . . . . .

Thank you for sending me the manuscript with the proof of Fermat’s Last Theorem. The first error is on page ... in line... . Because of it, the entire proof loses its validity.
Professor E. M. Landau

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

In the 1980s, Samuel Wagstaff raised the limit to 25,000, and in the 1990s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if you subtract even a trillion trillion from infinity, it will not become smaller. Mathematicians are not convinced by statistics. To prove the Great Theorem meant to prove it for ALL n going to infinity.

In 1954, two young Japanese mathematician friends began researching modular forms. These forms generate series of numbers, each with its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. No connection has ever been found between such different objects.

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

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

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

Having learned about Ken Ribet's findings, Wiles plunged headlong into proving the Taniyama-Shimura hypothesis. He decided to work in complete isolation and secrecy. “I realized that everything that had anything to do with Fermat’s Last Theorem arouses too much interest... Too many spectators obviously interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.

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

While the hype continued in the press, serious work began to verify the evidence. Every piece of evidence must be carefully examined before the evidence can be considered rigorous and accurate. Wiles spent a restless summer waiting for feedback from reviewers, hoping that he would be able to win their approval. At the end of August, experts found the judgment to be insufficiently substantiated.

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

“...half a minute after the start of the festive dinner on the occasion of her birthday, I presented Nadya with the manuscript of the complete proof” (Andrew Wales). Have I not yet said that mathematicians are strange people?

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

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

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

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