You may not know as many people in the world as you have never heard of Fermat's Last Theorem- Perhaps this is one mathematical problem that has gained such wide popularity and become a true legend. You can guess about it in many books and films, and the main context behind all the mysteries is impossibility of finishing the theorem.

So, this theorem has already been known and in the popular sense has become an “idol”, which is worshiped by amateur and professional mathematicians, but few people know about those whose proof was found, and this became the case already distant 1995. Let's talk about everything in order.

Also, Fermat's last theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician P'ierome Farm, is very simple in its essence and understands any people from the middle world. Let us say that the formula a n + b n = c n does not have natural (that is, not shotgun) solutions for n > 2. In fact, everything is simple and logical, but most mathematicians and simple amateurs have been struggling with the solution for three and a half centuries.

Fermat himself claimed that he had seen a very simple and concise proof of his theory, but until now no documented evidence of this fact has been found. That’s why it’s immediately important that you yourself Fermat was never able to find a definitive solution to his theorem, wanting to write a letter from your pen Viyshov private proof n = 4.

After Fermat, such great minds worked on the joke and proof as Leonard Yeyler(1770 was assigned a solution for n = 3), Adrian Legendre and Johann Dirichle(as early as 1825, a proof for n = 5 was found), Gabriel Lame(what is the best proof for n = 7) and many others. Until the mid-80s of the last century, it became clear that the light was now falling on the path of residual

Fermat's Great Theorem, until 1993, mathematicians began to admire and believe that Trivik's saga of trying to prove Fermat's remaining theorem was practically over.

1993 Roku English mathematician Andrew Wiles presenting your light proof of Fermat's Last Theorem, the work over which has been troubling over these fates. But it turned out that she decided to take revenge on a harsh punishment, even if it was done correctly. Wiles, without giving up, called for help from a well-known number theory scientist, Richard Taylor, and in 1994 they published corrections and additions to the proof of the theorem. What is most important is that this work borrowed as much as 130 (!) sums from the mathematical journal "Annals of Mathematics". However, the story didn’t end there - the final point was set only to the present day, 1995, as the most residual and “ideal”, from a mathematical point of view, version of the proof.

Almost an hour has passed since then, but everyone still has a clear idea about the inseparability of Fermat’s Last Theorem. Let those who know about the discovery of the proof continue the work in this direction - few people know that the Great Theorem will require a solution to 130 pages! Therefore, even rich mathematicians (most importantly amateurs, not professionals) were thrown into search of a simple and laconic proof, this path, which has led to everything, will not lead anywhere.

1

Ivliev Yu.A.

The article is devoted to a description of the principle of mathematical calculations made in the process of proving Fermat's Great Theorem at the end of the 20th century. It is revealed that the correct sense theorem is fulfilled, and it marks the development of a new axiomatic approach to the tracing of the stages of numbers and the natural series of numbers.

In 1995, an article was published, similar in size to a book, and it told about the proof of the famous Fermat's Great Theorem (LFT) (about the history of the theorem and attempts to bring it to wonders, for example). After this idea appeared without scientific articles and popular science books to promote this proof, in every day there was no explicit principle of mathematical atonement in the new one that crept in, not for the sake of it the author, and because of such wonderful optimism that having scoured the minds of mathematicians, who were engaged in the designated problem and its associated nutrition. The psychological aspects of this phenomenon were investigated by. There is also a detailed analysis of the agreement that was made, which is not of a private nature, but is a legacy of an incorrect understanding of the powers of the stages of whole numbers. As shown in , Fermat’s problem is rooted in a new axiomatic approach to the adoption of these authorities, which has not yet stagnated in modern science. Ale on this path has become a pardoned proof, which has given the number theory fakhists hybna guidelines and that the investigators of the Fermat problem have a direct and adequate solution. This robot is dedicated to this problem.

1. Anatomy of a pardon allowed under the hour of proof by the WTF

In the process of long and tedious mercury of the first hardness, Fermat was reformulated in terms of the formation of the diophantine level of the p-th stage with elliptic curves of the 3rd order (div. Theorems 0.4 and 0.5 c). Such a statement forced the authors of the actually collective proof to voice those that their method and mixing lead to a residual increase in Fermat’s problem (we would like to remind you that the WTF has a lot of known proofs for many stages of integers right up to the 90s of the last century). The method of this review is to establish the mathematical incorrectness of the stated statement and, as a result of the analysis, to find a principled compromise in the proof presented by.

a) Why do you have a pardon?

Also, in the text, on p. 448 it is said that after the “nice idea” of G. Frey, the possibility of proving the WTF was revealed. 1984 rock G. Frey letting go i

K. Ribet later confirmed that the elliptic curve was transferred, which represents the purpose of Fermat’s solution,

y 2 = x(x + u p) (x - v p) (1)

You can’t but it’s modular. However, A. Wiles and R. Taylor proved that any non-stable eliptic curve defined over the field of rational numbers is modular. Having learned about the impossibility of all decisions, Fermat’s jealousy and, therefore, about the fairness of Fermat’s affirmation, as in the words of A. Wiles, it was written as Theorem 0.5: there is no jealousy

u p+ v p+ w p = 0 (2)

de u, v, w- rational numbers, the whole indicator p ≥ 3; then (2) it is indicated only accordingly uvw = 0 .

Now, perhaps, we should turn back and critically evaluate why the curve (1) was a priori perceived as eliptical and what a real connection it is with Fermat’s work. In this regard, A. Wiles tries to consult Y. Hellegouarch, who knows how to construct a Fermat equation (virtually in whole numbers) a hypothetical curve of the 3rd order . In the opinion of G. Frey, I. Elleguarsh, without linking his curve with modular forms, used this method of removing the level (1) to further extend the proof of A. Wiles.

Let's start reporting on robots. The author carries out his research in terms of design geometry. The simplest actions of this designation and leading them to the appearance of, we know, a curve

Y 2 = X(X - β p)(X + γ p) (3)

a defiant level is established

x p+ y p+ z p = 0 (4)

de x, y, z- unknown numbers, p - the whole indicator of (2), and the solution of the Diophantine equation (4) α p, β p, γ p are used to record the curve (3).

Now, in order to see that the curve is eliptic to the 3rd order, it is necessary to look at the variables X and Y (3) on the Euclidean plane. For which the following rule of arithmetic of elliptic curves is known: if there are two rational points on a cubic algebra curve and a line that passes through these points intertwines this curve at one point, then the rest - a rational point. Hypothetical alignment (4) formally is the law of folding points onto straight lines. How to replace replacement parts x p = A, y p = B, z p = C and direct the curve in this manner straight along the X axis in (3), then the 3rd stage curve will be drawn at three points: (X = 0, Y = 0), (X = β p, Y = 0), (X = - γ p , Y = 0), which is shown in the curve entry (3) and in a similar entry (1). However, which curve (3) or (1) is truly eliptical? Obviously, no, because the sections of straight Euclidean with a folded point on it are taken on a nonlinear scale.

Rotating to the linear coordinate systems of the Euclidean space, we can eliminate the substitution of (1) and (3) formulas, even similar to the formulas for elliptic curves. For example, (1) can be in an offensive 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 establishment of the WTF seems unlawful. Regardless of those that (1) satisfies certain criteria for the class of elliptic curves, it still does not satisfy the most important criterion, which is the level of the 3rd level in the linear coordinate system.

b) Classification of milk

So, once again let’s go back to the beginning and simply look at how to get to the bottom of the truth of the WTF. First of all, it is transferred that the song is based on the solution of the Farm in positive integers. In another way, the solution is sufficiently inserted into the form of algebra of a certain type (a flat curve of the 3rd degree) in an assumption, which in this way eliminates the eliptic curves from appearing (another unconfirmed assumption). Thirdly, other methods can be used to ensure that a particular curve is non-modular, therefore, there is no difference. The result is clear: the whole decision is made on the Farm and, therefore, the WTF is correct.

These markings have one weak point, which after a detailed check appears to be weak. This remark is made at another stage of the proof process, when it is transferred that the hypothetical solution of Fermat’s equation is simultaneously solved by the solution of level 3 algebra, which describes an eliptic curve of the same kind. By itself, the assumption would have been justified, as if the curve of truth was meant to be elliptical. However, as can be seen from paragraph 1a), this curve is presented in non-linear coordinates, which makes it “illusory”, then. The linear topological space really doesn’t matter.

Now we must clearly classify the solution we have found. The point is that, as an argument to prove, those that need to be brought to the point are presented. In classical logic, this pardon is known as “poroche kolo.” In this case, the purpose of the decision is to construct the farm (perhaps quite unambiguously) with a fictitious, inconceivable elliptical curve, and then all the pathos of further delusions is used to convey what exactly it was A curve of this type is typical, and is drawn from hypothetical solutions, but the comparison is impossible.

How did it happen that such an elementary mistake was missed in serious mathematical work? Chantly, it happened through those who earlier mathematicians did not understand “illusory” geometric figures of the intended type. To be fair, who could have been caught, for example, by a fictitious colo, removed from the equal Farm by replacing the changeable ones x n/2 = A, y n/2 = B, z n/2 = C? Even the equation C 2 = A 2 + B 2 does not solve any problems for x, y, z and n ≥ 3. For nonlinear coordinate axes X and Y, the same formula was described, which looks very similar to the standard form:

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

where A and B are not changeable, but specific numbers, referred to as substitutes. If the numbers A and B are given a primary appearance, which is consistent with their static character, then the heterogeneity of values ​​among the partners on the right side of the equation immediately comes into view. This sign helps to show the illusion in action and move from non-linear coordinates to linear ones. On the other hand, if we look at the numbers as operators when they are equal to each other, as for example (1), then these and other functions will be the same values, then. guilty mothers, however, steps.

This understanding of the steps of numbers as operators also allows us to understand that the composition of the Fermat equation with an illusory eliptic curve is not unambiguous. Take, for example, one of the congeners on the right side of (5) and expand it into p linear congeners, introducing a complex number r such that r p = 1 (div. for example):

ξ p + u p = (ξ + u)(ξ + r u)(ξ + r 2 u)...(ξ + r p-1 u) (6)

This form (5) can be seen as being laid out in simple terms as multipliers of complex numbers on the base of algebraic identity (6), and the unity of such a breakdown can stand under the principle of nutrition, as was recently shown by Kummer.

2. Visnovki

From the previous analysis, it is clear that the so-called arithmetic of elliptic curves cannot shed light on those that require a proof of the WTF. After Fermat’s work, before the speech, taken by the epigrapher to this article, it began to feel like a historical heat and hoax. However, in reality it turns out that it was not Fermat who was on fire, but the fachians who gathered at the mathematical symposium in Oberwolfass in Germany in 1984, at which G. Frey voiced his interesting idea. The legacy of such a careless statement led mathematics to the gap between its loss of marital trust, which is well described and why there is no need to put before the science of nutrition the similarities of scientific attitudes before marriage. Fermat’s equation with the Frey curve (1) is the “lock” of Wiles’s entire proof based on Fermat’s theorem, and since there is no similarity between the Fermat curve and modular elliptical curves, there is no proof.

There are still a lot of Internet reports about those who apparently mathematicians decided to study Wiles’s proof of Fermat’s theorem, who came up with the idea of ​​​​justifying the view of the “minimal” by rearranging entire points at Euclidean space. However, no innovations can cover the classical results already obtained by mankind in mathematics, except for the fact that if we want to avoid an ordinal number with its analogue, we cannot replace it in opera this is the equalization of numbers among themselves, and It inevitably follows that Frey’s curve (1) will cease to be an eliptic cord, then. I don’t mean it for the reasons.

REFERENCES:

1. Ivliev Yu.A. Reconstruction of the native proof of Fermat’s Last Theorem – Scientific journal (section “Mathematics”). Kviten 2006 No. 7 (167) p.3-9, div. also Pratsі Lugansk branch of the International Academy of Information Technology. Ministry of Education and Science of Ukraine. Skhidnoukrainsk National University named after. V.Dal. 2006 r. No. 2 (13) p.19-25.
2. Ivliev Yu.A. The biggest scientific scam of the 20th century: “proof” of Fermat’s last theorem – Natural and technical sciences (section “History and methodology of mathematics”). Serpen 2007 r. No. 4 (30) p.34-48.
3. Edwards G. (Edwards H.M.) Fermat's last theorem. Genetic introduction to the theory of number algebra. Prov. from English per ed. B.F.Skubenko. M: Svit 1980, 484 p.
4. Hellegouarch Y. Points d´ordre 2p h sur les courbes elliptiques – Acta Arithmetica. 1975 XXVI p.253-263.
5. 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 mailing

Ivliev Yu.A. WILES'S PROOF OF FERMA'S LAST THEOREM // Fundamental Research. - 2008. - No. 3. - P. 13-16;
URL: http://fundamental-research.ru/ru/article/view?id=2763 (date of publication: 03/03/2020). We would like to present to you the magazines that are available at the Academy of Natural Sciences

The Abel Prize in 2016 goes to Andrew Wiles for his proof of the Taniyami-Shimuri conjecture for non-stable elliptic curves and the subsequent proof of Fermat's Last Theorem. At this time, the premium amounts to 6 million Norwegian kroner, or 50 million rubles. According to Wiles, the award of the prize became “a complete disappointment” for him.

Fermat's theorem, proven more than 20 years ago, still attracts the respect of mathematicians. Partly this is connected with these formulas, which is reasonable to tell the student: to show that for natural numbers n>2 there are no such three integers of non-zero numbers that a n + b n = c n . This Wislev P'ier Fermat wrote in the margins of Diophantus's "Arithmetic" with a miraculous signature: "I know the miraculous proof of [whose affirmation], but the margins of the book are too much for him." In addition to most mathematical stories, this is a reference.

The award ceremony is a wonderful reward for telling ten stories related to Fermat’s theorem.

1.

Before Andrew Wiles developed Fermat's theorem, it was more correctly called a conjecture, then Fermat's conjecture. On the right, the theorem has already been proven. However, I feel like this name has stuck to this firmament.

2.

Since Fermat’s theorem is for the domain n = 2, then such a comparison has an infinitely rich solution. These solutions are called “Pythagorean triplets”. This name was taken away from what the straight-cut thorns indicate, the sides of which are themselves expressed by such sets of numbers. You can generate Pythagorean triplets using the following three formulas (m 2 - n 2 , 2mn, m 2 + n 2). This formula requires different values ​​of m and n, and as a result we get the triplets we need. The main thing here, however, is that the numbers will be greater than zero - they cannot be expressed as negative numbers.

Before speaking, it is easy to note that if all the numbers in the Pythagorean triple are multiplied by a non-zero number, a new Pythagorean triple is obtained. Therefore, it is reasonable to add triplets, for which three numbers in the totality do not have a strong partner. The scheme we have described allows us to eliminate all such triplets - but this is by no means a simple result.

3.

On January 1, 1847, at a meeting of the Paris Academy of Sciences, two mathematicians - Gabriel Lame and Augustin Cauchy - announced that they were on the verge of proving the miracle theorem. They ruled the race, publishing little pieces of evidence. Most academicians rooted for Lamy, leaving Cauchy to be a self-righteous, intolerant religious fanatic (and, obviously, an absolutely brilliant mathematician behind the madness). Prote, the match was not destined to end - through his friend Joseph Liouville, the German mathematician Ernst Kummer informed the academicians that in the proofs of Cauchie and Lamie there was one and the same mercy.

The school has learned that the decomposition of numbers into simple multipliers is one. It was important for mathematicians to admire the calculation of whole numbers in a complex manner, so that power - unity - is preserved. However, this is not the case.

So, if you can only see m + i n, then the layout is one. Such numbers are called Gaussian. But for the work of Lamy and Koshy, it was necessary to sort the paper into multipliers in the cyclotomic fields. These, for example, are numbers in which m and n are rational, and i satisfies the power i^k=1.

4.

Fermat's theorem for n = 3 has a very geometric meaning. It is clear that we have a lot of small cubes. Let us take two great cubes from them. In this case, obviously, the sides will be whole numbers. Is it possible to find two great cubes, so that, having taken them out of the warehouse of fractional cubes, we could collect one great cube from them? Fermat's theorem seems to say that it is impossible to earn money like this. It’s funny that if you supply the same food for three cubes, then the evidence is solid. For example, this is the axis of four numbers, discovered by the wonderful mathematician Srinivas Ramanujan:

3 3 + 4 3 + 5 3 = 6 3

5.

In the history of Fermat's theorem, Leonard Euler appeared. No one was able to complete the assertion (or even approach the proof), but instead formulated a hypothesis about those who are equal

x 4 + y 4 + z 4 = u 4

There is no solution to whole numbers. All attempts to find solutions to such a head-on approach turned out to be fruitless. Only in 1988, Naum Elkies from Harvard managed to find a counter-butt. The axis looks like this:

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

Ask the formula to guess in a quiet numerical experiment. As a rule, in mathematics it looks like this: a simple formula. A mathematician verifies this formula using simple hypotheses, determines the truth and formulates a hypothesis. Then you (usually any graduate or student) write a program to verify that the formula is correct for reaching large numbers that cannot be touched with your hands (about one such experiment with simple numbers). This is not a proof, of course, but it’s a miracle to declare a hypothesis. All this is based on a reasonable assumption, that since there is a counter-attack to any reasonable formula, then we know how much it will take.

Euler's hypothesis suggests that life is much more diverse than our fantasies: the first counter-attack can be so great.

6.

In fact, it is clear that Andrew Wiles did not attempt to complete Fermat’s theorem - in a complex work called the Taniyami-Shimuri conjecture. Mathematics has two miraculous classes of objects. The first is called modular forms and is essentially the function of Lobachevsky’s space. These functions do not change when the surface itself changes. The other is called “elliptic curves” and “curves”, which are defined by the third-degree levels on the complex plane. Objects are even more popular in number theory.

In the 50s of the last century, two talented mathematicians Yutaka Taniyama and Goro Shimura met in the library of Tokyo University. At that time, there was no special mathematics at the university: it simply had not reappeared after the war. As a result, we have been working on old friends and studying at seminars ideas that in Europe and the USA were considered most important and not particularly relevant. Taniyama and Shimura themselves showed that there is a distinct similarity between modular forms and eliptic functions.

They verified their hypothesis on several simple crooked classes. It turned out that she was working. The stench of the stench was released, so that this connection is forever. This is how the Taniyami-Shimuri hypothesis emerged, and three years later Taniyama laid hands on himself. In 1984, the German mathematician Gerhard Frey showed that since Fermat's theorem is false, then the Taniyami-Shimuri conjecture is also false. It was hoped that the one who had proven this hypothesis would prove the theorem. Having earned it myself - though not entirely from the ignorant view - Wiles.

7.

Wiles spent everything he could to confirm his hypothesis. And in the hour of re-verification, the reviewers found a hint in it, which “hammered in” most of the evidence, starting to work again. One of the reviewers of the name, Richard Taylor, took a bet on Wiles's face. While the stinks were going on, it became known that Elkies, the same one who knew the counter-attack to Euler’s hypothesis, knew the counter-attack to Fermat’s theorem (later it turned out that this would be pepper heat). Wiles fell into depression and did not want to continue chewing - the door to proof was never closed. Taylor convinced Wiles to fight for another month.

It was a miracle and before the end of the summer mathematicians were able to make a breakthrough - this is how the work "Modular elliptic curves and Fermat's last theorem" by Andrew Wiles (pdf) and "Keltz-theoretic powers of Hecke's algebras" came to light. Charda Taylor and Andrew Wiles. This is already the correct proof. Published in 1995.

8.

In 1908, mathematician Paul Wolfskel died in Darmstadt. After depriving himself of the commandment that he gave to the mathematical partnership 99 years in order to know the proof of Fermat’s Last Theorem. The author of the proof is to take away 100 thousand marks (the author of the counter-example, to the point, without taking away anything). Because of the widespread legend, Wolfskehl’s mathematicians were prompted to make such a gift. This is how Simon Singh describes the legend in his book “Fermat’s Last Theorem”:

The story begins with Wolfskehl consuming a hot woman, the specificity of which has never been established. Much to the pity for Wolfskel, the mysterious woman threw him away. He fell into such deep discord that he decided to commit suicide. Wolfskel was a passionate person, but not an impulsive one, and began to detail his death in every detail. He recognized the date of his suicide and decided to shoot himself in the head with the first blow of the anniversary exactly the same day. During the rest of the day, Wolfskehl decided to organize his documents, which had gone miraculously, and on the remaining day, he said a commandment and wrote letters to close friends and relatives.

Wolfskel worked with such diligence that, having completed all his research before midnight, in order to fill in the yearbook that he had lost, he went to the library and began to look through mathematical journals. Recently, Kummer’s classic article was highlighted, in which he explained why Kosha and Lama’s misfortunes were recognized. Kummer's work preceded the most significant mathematical publications of his century and was most suitable for reading mathematicians who were planning to commit suicide. Wolfskel respectfully, row by row, quilted by Kummer's tabs. Unbelievably, Wolfskehl realized that he had discovered a clearing: the author had taken care of himself and did not waste the entire crop in his martyrdoms. Wolfskel began to chatter, and in truth I was able to reveal a serious clearing, where Kummer’s splintering had been lined. Once a clearing had been discovered, there was a chance that Fermat's Last Theorem could be developed much more simply, no matter who.

Wolfskehl sat at the table, carefully analyzed the “invaluable” part of Kummer’s merchandising and began to throw out a mini-proof that could either support Kummer’s work, or demonstrate the gentleness of the attitude he accepted and, as an inheritance Come on, let go of all your arguments. Wolfskel completed his calculations for Svitanka. The bad news (from the point of view of mathematics) was that Kummer's proof had become complete, and Fermat's Last Theorem, as before, had become inaccessible. Alas, there were good news: the hour of suicide had passed, and Wolfskel was so proud that I managed to reveal and fill the gap in the work of the great Ernest Kummer, so that his pain and troubles resolved themselves. Mathematics has turned you into life.

However, there is an alternative version. Together with her, Wolfskell took up mathematics (and, above all, Fermat’s theorem) through progressive sclerosis, which led him to take up his favorite field - being a doctor. And depriving the mathematicians of their money so as not to deprive their squad, who simply hate until the end of their lives.

9.

Attempts to prove Fermat’s theorem using elementary methods resulted in the emergence of a whole class of wonderful people under the name of “Fermatists.” They did this in order to collect a great amount of evidence and did not at all give in to witnesses if they found a compromise in this evidence.

At the Faculty of Mechanics and Mathematics of the MDU there is a legendary character called Dobretsov. He collected evidence from various departments and fought with them, penetrating the mechanical engineering department. It was all about finding the victim. I think I came across a young graduate student (academician Novikov). In his honesty, he began to respectfully read one hundred papers, which Dobretsov slipped into his words, saying, the axis of proof. After the meal, “Wasp milk...” Dobretsov took the glass and snorted it into his briefcase. From another briefcase (so, he was walking around with two briefcases) he pulled out another hundred, sighed and said: “Well, then I’m surprised at option 7 B.”

Before speaking, most of such evidence begins with the phrase “Let us transfer one of the contributions to the right part of equality and decompose it into multipliers.”

10.

The story about the theorem would not be complete without the wonderful film “The Mathematician and the Devil.”

Vipravlenya

In section 7 of the article it was immediately stated that Naum Elkies knew the counter-application to Fermat’s theorem, which soon appeared in mercy. This is wrong: the information about the counter-butt was a shocking blast. Please excuse me for inaccuracy.

Andriy Konyaev

Great Fermat Theorem Singh Simon

"Has Fermat's Last Theorem been proven?"

It was only a matter of time before proving the Taniyami-Shimuri conjecture, but the strategy developed by Wiles was a brilliant mathematical breakthrough, a result that merited publication. However, through the habit of the war imposed by Wiles on himself, there was no information about the final result of the decision to the world and there was no sign of anyone who could make such a significant breakthrough.

Wiles speaks of his philosophical position before any potential rival: “No one wants to waste time proving something and discovering that someone else managed to know the proof of something many years earlier. Well, it’s not surprising that as soon as I tried to figure out the problem, because, in essence, I was indifferent, I was not even afraid of the superniks. I just couldn’t imagine that I would be less inclined to think about the idea that I would bring to the proof.”

On February 8, 1988, Wiles, in shock, saw headlines in large font on the front of the newspapers, saying: “Fermat’s great theorem has been proven.” The Washington Post and New York Times reported that 38-year-old Yoichi Miyaoka of Tokyo Metropolitan University had identified the world's most important mathematical problem. While Miya had not yet published his proof, he had already announced his progress at a seminar at the Max Planck Institute for Mathematics in Bonn. Don Tsagir, who was present at Miyaoka’s speech, expressed the optimism of mathematical prowess in the following words: “Miyaoka’s proof is extremely compelling, and mathematicians respect that there is a high level of I have to appear correct. There is still no certainty, but for now the proof looks very encouraging.”

Coming from a report at a seminar in Bonny, there are some thoughts about his approach to the problem, as seen from a completely different, algebraic-geometric point of view. The remaining decades of geometry have reached a deep and subtle understanding of mathematical objects, the surface, and the powers of the surface. In the 70s, the Russian mathematician S. Arakelov tried to establish parallels between the problems of geometry algebra and the problems of number theory. As a direct result of Langlands's program, mathematicians realized that unsolved problems in number theory could be solved, along with other problems in geometry, which were also left unsolved. This program came under the name of the philosophy of parallelism. Those geometries of algebra, which dealt with the problems of number theory, were called “arithmetic algebraic geometries”. In 1983, they announced their first significant victory, when Gerd Faltings from the Princeton Institute of Greater Research had made a significant contribution to Fermat's fundamental theorem. Let's guess what, behind the strongholds of the Farm, Rivnyanya

at n There are no solutions for whole numbers greater than 2. Faltings believed that he was able to poke his head into the proof of Fermat's Last Theorem through the additional development of geometric surfaces associated with different values n. Surface, knitted with burlap Truss for different values n, they look like one another, but they are hiding one secret power - their ears are completely open, or, simply apparent, holes. These surfaces are as diverse as graphics of modular forms. Two-dimensional cuts of two surfaces are shown in Fig. 23. The surfaces, knitted from Truss ropes, look similar. The greater the value n in the plain, there are more trees on the surface.

Small 23. These two surfaces were drawn using the additional computer program “Mathematica”. The skin of them represents a geometric point that satisfies the skin. x n + y n = z n(for the surface of the earth n=3, right-handed on the surface n=5). Zminni xі y here we take a more complex approach

Faltings was able to conclude that, if fragments of such surfaces were ever tossed around a number of trees, the Rivne Farm associated with them could lead to even more final, impersonal decisions for whole numbers. The number of decisions could be anything from zero, as Fermat passed on, to a million or a billion. Thus, Faltings did not agree with Fermat's Last Theorem, but he decided to throw up the possibility of Fermat's theory of endlessly rich solutions.

Five fates later, Miyaoka told me that I had managed to get through one more krok. I was then twenty years old. Miyaoka formulated the hypothesis of some kind of inequality. It became clear that proving this geometric hypothesis would mean proving that Fermat’s number is not just primer, but zero. Miyaki's approach was similar to Wiles's in that they both attempted to develop Fermat's Last Theorem by linking it to a fundamental hypothesis in another branch of mathematics. Miyaoka had the geometry of algebra, for Wiles the route to proof lay through eliptic curves and modular forms. Much to the pity of Wiles, he was still struggling with the proof of the Taniyama-Shimura hypothesis, when Miyaoka told him that he had the final proof of his hypothesis and, therefore, Fermat’s Last Theorem.

Two years after his appearance in Bonn, Miyaoka published five sides of the calculation, which formed the essence of his proof, and a thorough re-verification began. Scientists on number theory and geometry algebra in all corners of the world have published row after row of published calculations. After a few days, mathematicians revealed one super-accuracy in the proof, which could not help but raise concerns. One of the parts of Miyaoki’s work led to the solidification of number theory, which, when transferred to the mathematical geometry of algebra, resulted in a solidification that was consistent with the result that had been denied by many fates earlier. And although this did not necessarily mean the entire proof of Miyaoka, the revealed contradiction did not fit into the philosophy of parallelism between number theory and geometry.

Another two years later, Gerd Faltings, who broke the path of Miyaoke, spoke about those who had revealed the exact cause of the breakdown in parallelism that appears - a gap in the world. The Japanese mathematician, who was a geometer, was absolutely brilliant in transferring his ideas to those familiar with the territory of number theory. An army of number theorists has made a vigorous effort to patch up the hole in Miyaoka's proof, aka. Two months after Miyaoka announced that there might be a final proof of Fermat's Last Theorem, the mathematical partnership came up with a one-piece development: Miyaoka's proof of adverbs for failure.

As if there was a lot of evidence that had not come to fruition, Miyaoke was able to discard many of the results. Other fragments of his proof were credited with the advanced additions of geometry to the theory of numbers, and in recent years other mathematicians used them to prove various theorems, but no one could achieve Fermat’s Last Theorem this way.

The buzz around Fermat's Great Theorem soon died down, and the newspapers ran short notices saying that the three-hundredth puzzle, as before, was becoming unsolved. On the wall of the New York subway station on Eighth Street appeared an inscription, no doubt inspired by press publications inspired by Fermat's Great Theorem: xn + yn = zn There is no solution. I know the truly amazing proof of this fact, but I can’t write it down here, because it’s too difficult for me.”

Section ten CROCODILE FARM The Stinks were driving along the fry road in old John's car, sitting in the back seats. Behind the kerm there was black water in a bright shirt with a chimerically cropped head. On the shaved skull hung bushes of coarse, like drittish, black hair, logic.

Preparation before the race. Alaska, Lindy Pletner's farm "Iditarod" - short dog sledding in Alaska. The length of the route is 1150 miles (1800 km). This is the world's favorite sled dog race. Start (urochisty) - 4 Bereznya 2000 from Anchorage. Start

Goat farm Robots entered the village. When we reached the village of Khomutets, hay was being prepared there and the pine needles from the freshly cut grass seemed to have leaked all over. Qiu

Letnya farm Straw, hand-made, folded into the grass; Insha, having signed her name on the parka, set fire to the fire of the green fire of Vodya in Koriti Kinsky. Nine jocks walk along a number of parallel lines, walking around the blue day. The trigger axis marveled at nothing

The farm was built in the Calm Sun with a dark red flower bowed down to the ground, rising in the setting of the sun, as night fell in the empty space, dimming the light, which disturbed the gaze. Silence fell on the farm without a breath, Before her hair was gone, It was beating over the cactus

What farm do you trust? On the 13th of 1958, all central Moscow, and subsequently regional newspapers published the decision of the Central Committee of the Communist Party of Ukraine “On the amends for the purchase of measles from collective farmers in the Zaporizhzhya region.” It was necessary to tell not about the whole region, but about two districts: Primorsky

Fermat's Problem In 1963, when he was more than ten years old, Andrew Wiles was already fascinated by mathematics. “At school, I loved to see the treasures, I took them home and picked up new ones from the workshop. Ale, best of all, from the legacy that I was subjected to, I discovered at the place

From the Pythagorean theorem to Fermat's Last Theorem About the Pythagorean theorem and the endless number of Pythagorean triplets were discussed in the book by E.T. Bella “The Great Problem” is the library book that won the respect of Andrew Wiles. And although the Pythagoreans reached the greatest possible

Mathematics after the proof of Fermat’s Great Theorem It’s not surprising that Wiles himself, in his own words, sensed the confusion: “The opportunity for the appearance of the meetings is already far away, but the lecture itself was crying out for me. I feel it. Working on the proof

Section 63 Old McLennon's Farm About a month after turning back to New York one leafy evening, the phone rang at the Lennons' apartment. Yoko answered the phone. A human voice with a Puerto Rican accent asked Yoko Ono. Pretending

Pontryagin's theorem At the same time at the Conservatory I started at the MDU, at the Faculty of Mechanics and Mathematics. Having successfully completed it, you will continue to spend many hours choosing a profession. Musicology prevailed, and as a result it won over my mathematical mind. One of these classmates

Theorem The theorem about the right of a religious association to rob a priest will require proof. It reads like this: “The Orthodox community is being created... under the spiritual leadership of the enshrined community and the blessing of the diocesan bishop priest.”

I. Farm (“Here, in front of the smoking residue...”) Here, in front of the smoking residue. One step - a broom. Kohannya - yak for rahunkom? - She took me to the barn. The grain is pricking, the chickens are cackling, the stumps are croaking with importance. And without size or censorship, worlds are formed in the mind. About Provençal noon

There are not many people in the world who have never heard of Fermat's Last Theorem - perhaps the only mathematical problem that has gained such wide popularity and become a true legend. You can guess about it in the impersonality of books and films, in which the main context behind all mysteries is the impossibility of completing the theorem.

So, this theorem has already been known and in the popular sense has become an “idol”, which is worshiped by amateur and professional mathematicians, but few people know about those whose proof was found, and this became the case already distant 1995. Let's talk about everything in order.

Also, Fermat’s last theorem (often called Fermat’s last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in its essence and understandable to any person from the middle world. It should be said that the formula a in step n + b in step n = c in step n does not have natural (that is, not shot) solutions for n > 2. In fact, everything is simple and logical, but most mathematicians and simple amateurs struggled with the search The decision is to pay three and a half cents.

Why is she so famous? Let's find out now...

How many theorems have not been completed, have not been completed, and have not yet been completed? The whole point here is that Fermat’s Last Theorem is the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Great Theorem is incredibly important, its formulation can be understood by 5th graders of secondary school, and the proof is something that not every professional mathematician can understand. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics, there is no problem that would be formulated so simply, but would remain undetected for so long. 2. Why is it lying there?

Let's start with Pythagorean pants. The formula is really simple - at first glance. As we know from childhood, “Pythagorean pants are equal on all sides.” The problem appears to be as simple as the fact that it was based on a mathematical statement, as everyone knows, - Pythagoras’ theorem: if any rectilinear triangle has a square on the hypotenuse, the ancient sum of squares on the legs.

In the 5th century BC Pythagoras fell asleep the Pythagorean brotherhood. The Pythagoreans, among other things, calculated as many as three to satisfy the equalities x²+y²=z². They realized that the Pythagorean triplets were infinitely rich, and they found hidden formulas for their use. Chantingly, the stench was noticeable on the three and higher steps. Having drunk too much and not going out, the Pythagoreans lost their taste. The members of the brotherhood were more philosophers and aesthetes, less mathematicians.

It’s easy to choose impersonal numbers that miraculously satisfy the equalities x²+y²=z²

Starting with 3, 4, 5 – it’s true, the young school student realized that 9+16=25.

Abo 5, 12, 13: 25 + 144 = 169. Miraculous.

So it appears that there are none. This is where the tricky stuff begins. Simplicity - it seems that it is important to convey not the obviousness of what, but rather, the reality. If you need to convey that a decision has been made, it is possible and necessary to simply bring about a decision.

To make the reality more complex: for example, although it seems: such jealousy is not a solution. Put Yogo in Kalyuzha? easy: bam - and the axis is there, decision! (Bring a solution). І all, opponent of enemies. How can we achieve daily activity?

Say: “I don’t know such decisions”? Or maybe you’re playing a nasty prank? And the stench is so strong that the strained computer is still exhausted? The axis is foldable.

In practical terms, it can be shown this way: if you take two squares of similar sizes and divide them into single squares, then for the price of purchasing single squares you will get a third square (Fig. 2):


And let’s get to the third world (Fig. 3) – don’t go out. Reject cubes, otherwise you will lose claims:

And the axis of the mathematician of the 17th century, the Frenchman Pierre de Fermat, traced the underground level xn+yn=zn from buried treasures. I, solve by resolving: for n>2 there is no solution. Fermat's proof is irrevocably spent. Manuscripts are burning! He lost his respect in Diophantus’s “Arithmetic”: “I know a truly amazing proof of this proposition, but the fields here are too small to accommodate him.”

A theorem without proof is called a hypothesis. Ale Fermat's fame has been cemented, and he will never have mercy. The truth is that without depriving the evidence of any assertion, it was confirmed over the years. Before that, Fermat completed his thesis for n=4. So the hypothesis of the French mathematician went down to history as Fermat's Last Theorem.

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

Adrian Legendre and Johann Dirichlet (in 1825 they found a proof for n = 5), Gabriel Lamé (who found a proof for n = 7) and many others. Until the mid-80s of the last century, it became clear that the world would be on the way until the residual validity of Fermat’s Last Theorem, but only in 1993 mathematicians began to believe and believed that Trivic’s epic was in search of proof In the remaining theorem, Fermat is practically finished.

It is easy to see that it is enough to prove Fermat’s theorem only for simple n: 3, 5, 7, 11, 13, 17, ... For large n, the proof is no longer straightforward. There are countless numbers of prime numbers.

In 1825, a group of women who developed the method of Sophie Germain, female mathematicians, Dirichlet and Legendre, in one way or another, arrived at a theorem for n=5. In 1839, using this very method, the Frenchman Gabriel Lame showed the truth of the theorem for n=7. Step by step, the theorem was extended to all n less than a hundred.

Finally, the German mathematician Ernst Kummer showed in a brilliant manner that it was impossible to prove the theorem in reality using the methods of mathematics of the 19th century. The Prize of the French Academy of Sciences, awarded in 1847 for the proof of Fermat's theorem, remained unawarded.

In 1907, a wealthy German industrialist, Paul Wolfskel, was born through an undivided business and wanted to bring the world to life. As a true German, he recognized the date and hour of suicide: exactly the same night. On the last day, the family will command and write letters to friends and relatives. The service ended early in the afternoon. It should be said that Paul was interested in mathematics. With nothing to do, go to the library and start reading Kummer’s famous article. Unbelievably, it seemed to him that Kummer had made a peace on his way out. Wolfskehl became an olivian in the hands of the city. The night has passed and the morning has come. The gap in the evidence was filled. That same reason for committing suicide now looks absolutely mindless. Paul opened the farewell pages and rewrote the commandment.

Nezabar died of natural causes. The decline was not always awarded: 100,000 marks (over 1,000,000 sterling) were transferred to the Royal Scientific Association of Göttingen, as the same fate announced the holding of a competition for the Wo Prize fskelya. 100,000 marks were deposited into Fermat's theorem, which was proven. Not a penny was due for the formulation of the theorem...

Most professional mathematicians respected the effort to prove Fermat's Last Theorem without hope and were reluctant to spend an hour on such a busy task. Then the fans had a blast. Several years after the shock, an avalanche of “evidence” fell on the University of Göttingen. Professor E.M. Landau, who included the analysis of the evidence above, handed out cards to his students:

Shanovniy(a). . . . . . . .

Thank you for sending me a manuscript with a proof of Fermat's Last Theorem. The first pardon appears on the side. ... in a row... . Through it, the entire proof is drained of decorum.
Professor E. M. Landau

In 1963, Paul Cohen, building on Gödel’s ideas, arrived at the incoherence of one of Hilbert’s twenty-three problems—the continuum hypothesis. What if Fermat’s Great Theorem is also inextricable? True fanatics of the Great Theorem were not disappointed. The advent of computers gave mathematicians a new way of confirmation. After the Other Light War, groups of programmers and mathematicians brought Fermat's Last Theorem for all values ​​of n to 500, then to 1,000, and later to 10,000.

In the 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians declared that Fermat's Last Theorem is true for all values ​​of n up to 4 million. If you choose a trillion trillions from the lack of diversity, you will not become less. Mathematicians are not converted by statistics. To bring the Great Theorem to light meant to bring її ALL n, as in inconsistency.

In 1954, two young Japanese mathematician friends began researching modular forms. These forms give rise to rows of numbers, and the skin – its own row. Vipadkovo Taniyama aligned these rows with rows, which would give rise to elliptic rows. The stinks are gone! All modular forms are geometric objects, and eliptic forms are algebraic. No connection has ever been found between such different objects.

At the same time, friends, after careful verification, came up with a hypothesis: the cutaneous elliptical skin has a twin – a modular form, and so on. This hypothesis itself became the foundation of the whole directly in mathematics, but until the Taniyami-Shimuri hypothesis was completed, the whole world could collapse in any way.

In 1984, Gerhard Frey showed that Fermat's solution, as it turns out, can be included in an elliptical equation. Two fates later, Professor Ken Ribet Dov, that this hypothetical equal cannot be the mother of a twin in the modular world. From now on, Fermat's Last Theorem was inextricably linked with the Taniyami-Shimuri conjecture. Having established that no matter how eliptic a curve is modular, it is important to note that there is no eliptic equation with Fermat’s solutions, and Fermat’s Last Theorem was immediately completed. After thirty years it was not possible to complete the Taniyami-Shimuri hypothesis, and less hope of success was lost.

In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. Once you know about the Great Theorem, you realize that you cannot make sense of it. As a schoolboy, a student, a graduate student, I prepared myself for such a task.

Having learned about Ken Ribet's ideas, Wiles set out to prove the Taniyami-Shimuri hypothesis. You prefer to use complete isolation and secrecy. “I understand that everything that may be related to Fermat’s Last Theorem arouses great interest... There are too many forward-looking people who respect what has been achieved.” This hard work bore fruit, Wiles found that he had completed the proof of the Taniyami-Shimuri conjecture.

In 1993, the English mathematician Andrew Wiles presented his light proof of Fermat's Last Theorem (Wiles read his sensational proof at a conference at the Institute of Sir Isaac Newton in Cambridge), a work on the current problem c.

While the news continued in the press, serious work began on verifying the evidence. Kozhen fragment is guilty of proof buti retelno vivcheny persh nizh proof mozhe buti vyznany suvorim ta toch. Wiles spent a turbulent summer in the wake of reviewers, confident that he would be able to retract the praise. For example, a number of experts revealed that the judgment was insufficiently grounded.

It turned out that the decision was made to take revenge on the harsh punishment, even if it was done right. Wiles did not give up, calling for help from Richard Taylor, a well-known fachian in number theory, and already in 1994 they published corrections and additional proofs of the theorem. What is most important is that this work borrowed as much as 130 (!) sums from the mathematical journal "Annals of Mathematics". However, the story didn’t end there - the final point was put before the present day, 1995, as the most residual and “ideal”, from a mathematical point of view, version of the proof.

“... just after the beginning of the Yuletide dinner on the eve of the nation’s day, I presented Nadya with a manuscript of complete proof” (Andrew Wals). Have I not yet said that mathematicians are wonderful people?

This time there was no doubt about the proof. Two articles were subject to theoretical analysis and were published in the journal “Annals of Mathematics” in 1995.

Almost an hour has passed since that moment, but the mind still has a clear idea about the inseparability of Fermat’s Last Theorem. Let those who know about the discovery of the proof continue the work in this direction - few people know that the Great Theorem will require more than 130 pages!

Therefore, even rich mathematicians (mostly amateurs, not professionals) were thrown into search of a simple and laconic proof, along this path, which has led to everything, will not lead anywhere...

Dzherelo