The farm equation is currently undecidable. Fermat's Last Theorem: proof of Wiles and Perelman, formulas, calculation rules and complete proof of the theorem

16.12.2019

Judging by the popularity of the query "Fermat's theorem - short proof ", this mathematical problem really interests many. This theorem was first expressed by Pierre de Fermat in 1637 at the edge of a copy of Arithmetic, where he claimed that he had its solution, it was too large to fit on the edge.

The first successful proof was published in 1995 - it was a complete proof of Fermat's theorem by Andrew Wiles. It has been described as "overwhelming progress" and led Wiles to the 2016 Abel Prize. Described relatively briefly, the proof of Fermat's theorem also proved most modularity theorems and opened up new approaches to numerous other problems and effective methods the rise of modularity. These accomplishments propelled mathematics 100 years forward. The proof of Fermat's little theorem today is not something out of the ordinary.

An unsolved problem stimulated the development of algebraic number theory in the 19th century and the search for a proof of the modularity theorem in the 20th century. This is one of the most notable theorems in the history of mathematics, and until the complete proof of Fermat's theorem by division, it was in the Guinness Book of Records as “the most difficult mathematical problem”, one of the features of which is that it has the largest number of failed proofs.

Historical reference

The Pythagorean equation x 2 + y 2 \u003d z 2 has an infinite number of positive integer solutions for x, y and z. These solutions are known as the Pythagorean trinity. In about 1637, Fermat wrote at the edge of the book that the more general equation an + bn \u003d cn has no natural solutions if n is an integer greater than 2. Although Fermat himself claimed to have a solution to his problem, he did not leave no details about her proof. The elementary proof of Fermat's theorem, stated by its creator, was rather his boastful invention. The book of the great French mathematician was discovered 30 years after his death. This equation, called "Fermat's Last Theorem", remained unsolved in mathematics for three and a half centuries.

The theorem eventually became one of the most notable unsolved problems in mathematics. Attempts to prove this caused a significant development in number theory, and over time, Fermat's last theorem became known as an unsolved problem in mathematics.

A brief history of the evidence

If n \u003d 4, which was proved by Fermat himself, it suffices to prove the theorem for the indices n, which are prime numbers. Over the next two centuries (1637-1839), the conjecture was only proven for primes 3, 5, and 7, although Sophie Germain updated and proved an approach that was relevant to the entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, with the result that the irregular primes were parsed individually. Building on Kummer's work and using sophisticated computer science, other mathematicians were able to extend the solution of the theorem, with the goal of covering all major indicators to four million, but the proof for all exponents was still not available (meaning that mathematicians usually considered the solution impossible, extremely difficult, or unattainable with modern knowledge).

Shimura and Taniyama's work

In 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected there was a connection between elliptic curves and modular shapes, two completely different areas of mathematics. Known at the time as the Taniyama-Shimura-Weil conjecture and (ultimately) as the modularity theorem, it existed on its own, with no apparent connection with Fermat's last theorem. It itself was widely regarded as an important mathematical theorem, but it was considered (like Fermat's theorem) impossible to prove. At the same time, the proof of the great Fermat's theorem (by the method of division and the use of complex mathematical formulas) was carried out only half a century later.

In 1984 Gerhard Frey noticed an obvious connection between these two previously unrelated and unresolved issues. Full confirmation that the two theorems were closely related was published in 1986 by Ken Ribet, who based on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture." Simply put, these papers by Frey, Serre, and Ribe showed that if the modularity theorem could be proved, at least for a semistable class of elliptic curves, then the proof of Fermat's last theorem would also sooner or later be discovered. Any solution that might contradict Fermat's last theorem can also be used to contradict the modularity theorem. Therefore, if the modularity theorem turned out to be true, then by definition there cannot be a solution that contradicts Fermat's last theorem, which means that it had to be proved soon.

Although both theorems were tricky problems for mathematics, considered unsolvable, the work of the two Japanese was the first suggestion of how Fermat's last theorem could be continued and proved for all numbers, not just a few. Important for the researchers who chose the research topic was the fact that, unlike Fermat's last theorem, the modularity theorem was the main active area of \u200b\u200bresearch for which a proof was developed, and not just a historical oddity, so the time spent on its work could be justified from a professional point of view. However, the general consensus was that the solution to the Taniyama-Shimura hypothesis turned out to be inappropriate.

Fermat's Last Theorem: Wiles' proof

Having learned that Ribet had proved the correctness of Frey's theory, the English mathematician Andrew Wiles, who was interested in Fermat's last theorem from childhood and who had experience with elliptic curves and adjacent domains, decided to try to prove the Taniyama-Shimura conjecture as a way to prove Fermat's last theorem. In 1993, six years after announcing his goal, while secretly working on the problem of solving a theorem, Wyles succeeded in proving a related conjecture, which in turn would have helped him prove Fermat's last theorem. Wiles's document was enormous in size and scope.

The flaw was discovered in one part of his original article during peer review and required another year of collaboration with Richard Taylor to jointly solve the theorem. As a result, Wiles' final proof of Fermat's theorem was not long in coming. In 1995, it was published on a much smaller scale than Wiles's previous mathematical work, clearly showing that he was not mistaken in his previous conclusions about the possibility of proving the theorem. Wiles' achievement was widely disseminated in the popular press and popularized in books and television programs. The rest of the Taniyama-Shimura-Weil conjecture, which was now proven and known as the modularity theorem, was subsequently proven by other mathematicians who based on Wiles' work between 1996 and 2001. For his achievement, Wiles has been honored and received numerous awards, including the 2016 Abel Prize.

Wiles' proof of Fermat's last theorem is a special case of the solution of the modularity theorem for elliptic curves. However, this is the most famous case of such a large-scale mathematical operation. Along with the solution of Ribe's theorem, the British mathematician also obtained a proof of Fermat's last theorem. Fermat's last theorem and modularity theorem were almost universally considered unprovable by modern mathematicians, but Andrew Wiles was able to prove to the entire scientific world that even pundits can be deluded.

Wiles first announced his discovery on Wednesday June 23, 1993 at a lecture in Cambridge entitled "Modular Shapes, Elliptic Curves and Galois Representations." However, in September 1993, it was found that his calculations contained an error. A year later, on September 19, 1994, in what he would call “the most important moment of his working life,” Wiles stumbled upon a revelation that allowed him to correct his problem solution to the point where it could satisfy the mathematical community.

Job characteristics

The proof of Fermat's theorem by Andrew Wiles uses many methods from algebraic geometry and number theory and has many ramifications in these areas of mathematics. He also uses the standard constructions of modern algebraic geometry such as the category of schemes and Iwasawa's theory, as well as other 20th century methods that were not available to Pierre Fermat.

The two pieces of evidence are 129 pages long and were written over seven years. John Coates described this discovery as one of the greatest achievements in number theory, and John Conway called it the main mathematical achievement of the 20th century. Wiles, to prove Fermat's last theorem by proving the modularity theorem for the particular case of semi-stable elliptic curves, developed effective methods the rise of modularity and opened up new approaches to numerous other problems. For solving Fermat's last theorem, he was knighted and received other awards. When it became known that Wiles had won the Abel Prize, the Norwegian Academy of Sciences described his achievement as "a delightful and rudimentary proof of Fermat's last theorem."

How it was

One of the people who analyzed Wiles's original manuscript with the solution to the theorem was Nick Katz. During his review, he asked the Briton a series of clarifying questions, which led Wiles to admit that his work clearly contains a gap. In one critical part of the proof, a mistake was made that gave an estimate for the order of a particular group: the Euler system used to extend the Kolyvagin and Flach method was incomplete. The mistake, however, did not render his work useless - every part of Wiles's work was very significant and innovative in itself, as were many of the developments and methods that he created in the course of his work, which affected only one part of the manuscript. However, this original paper, published in 1993, did not really have a proof of Fermat's Last Theorem.

Wiles spent nearly a year trying to re-solve the theorem - first alone and then in collaboration with his former student Richard Taylor - but it seemed to be in vain. By the end of 1993, rumors circulated that Wiles's proof had failed in verification, but how severe the failure was was not known. Mathematicians began to pressure Wiles to reveal the details of his work, whether it was completed or not, so that the broader mathematician community could explore and use whatever he was able to achieve. Instead of quickly correcting his mistake, Wiles only discovered additional complex aspects in the proof of Fermat's Last Theorem, and finally realized how difficult it is.

Wiles states that on the morning of September 19, 1994, he was on the verge of giving up and giving up and almost resigned himself to failing. He was ready to publish his unfinished work so that others could build on it and find where he was wrong. The English mathematician decided to give himself one last chance and analyzed the theorem one last time in order to try to understand the main reasons why his approach did not work, when he suddenly realized that the Kolyvagin-Flak approach would not work until he included Iwasawa's theory, making it work.

On October 6, Wiles asked three colleagues (including Faltins) to review his new work, and on October 24, 1994, he submitted two manuscripts - "Modular Elliptic Curves and Fermat's Last Theorem" and "Theoretical Properties of the Ring of Certain Hecke Algebras", of which Wiles co-wrote with Taylor and proved that certain conditions were met to justify the revised step in the main article.

These two articles were reviewed and finally published as a full-text edition in the May 1995 Annals of Mathematics. Andrew's new calculations were widely analyzed and eventually accepted by the scientific community. These papers established the modularity theorem for semistable elliptic curves - the last step towards the proof of Fermat's last theorem, 358 years after it was created.

History of the great problem

The solution to this theorem has been considered the biggest problem in mathematics for centuries. In 1816 and 1850, the French Academy of Sciences offered a prize for the general proof of Fermat's last theorem. In 1857, the Academy awarded 3000 francs and the gold medal to Kummer for his research on ideal numbers, although he did not apply for the prize. Another prize was offered to him in 1883 by the Brussels Academy.

Wolfskel Prize

In 1908, the German industrialist and amateur mathematician Paul Wolfskel bequeathed 100,000 gold marks (a large sum for that time) to the Academy of Sciences of Göttingen, so that this money would become a prize for the complete proof of the great Fermat's theorem. On June 27, 1908, the Academy published nine awards rules. Among other things, these rules required the proof to be published in a peer-reviewed journal. The prize was to be awarded only two years after publication. The competition was due to expire on September 13, 2007 - about a century after its start. On June 27, 1997, Wiles received Wolfshel's prize money, followed by another $ 50,000. In March 2016, he received € 600,000 from the Norwegian government as part of the Abel Prize for "a stunning proof of Fermat's last theorem using the modularity hypothesis for semi-stable elliptic curves, opening a new era in number theory." It was a world triumph for the humble Englishman.

Before Wiles's proof, Fermat's theorem, as mentioned earlier, was considered absolutely unsolvable for centuries. Thousands of false evidence in different time were presented to the Wolfskel committee, amounting to approximately 10 feet (3 meters) of correspondence. In the first year of the existence of the prize alone (1907-1908), 621 applications were filed with a claim to solve the theorem, although by the 1970s their number had decreased to about 3-4 applications per month. According to F. Schlichting, Wolfschel's reviewer, most of the evidence was based on elementary methods taught in schools, and was often presented as "people with technical education, but unsuccessful careers." According to the historian of mathematics Howard Aves, Fermat's last theorem set a kind of record - this is the theorem that received the most incorrect evidence.

Farm laurels went to the Japanese

As mentioned earlier, around 1955, the Japanese mathematicians Goro Shimura and Yutaka Taniyama discovered a possible connection between two apparently completely different branches of mathematics - elliptic curves and modular shapes. The resulting modularity theorem (at the time known as the Taniyama-Shimura conjecture) states that every elliptic curve is modular, which means that it can be associated with a unique modular shape.

The theory was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence to support the Japanese conclusions. As a result, the hypothesis was often called the Taniyama-Shimura-Weil hypothesis. It became part of the Langlands program, which is a list of important hypotheses to be proven in the future.

Even after serious scrutiny, the hypothesis was recognized by modern mathematicians as extremely difficult or, perhaps, impossible to prove. Now this very theorem is waiting for its Andrew Wiles, who could surprise the whole world with its solution.

Fermat's theorem: proof of Perelman

Despite the popular myth, the Russian mathematician Grigory Perelman, for all his genius, has nothing to do with Fermat's theorem. Which, however, does not in any way detract from his many services to the scientific community.

n\u003e 2 (\\ displaystyle n\u003e 2) the equation:

has no solutions in non-zero integers.

There is a narrower version of the formulation asserting that this equation has no natural solutions. However, it is obvious that if there is a solution for integers, then there is a solution in natural numbers. Indeed, let a, b, c (\\ displaystyle a, b, c) - integers giving the solution to Fermat's equation. If a n (\\ displaystyle n) even then | a | , | b | , | c | (\\ displaystyle | a |, | b |, | c |) will also be a solution, and if it is odd, then we transfer all degrees of negative values \u200b\u200bto another part of the equation, changing the sign. For example, if there were a solution to the equation a 3 + b 3 \u003d c 3 (\\ displaystyle a ^ (3) + b ^ (3) \u003d c ^ (3)) and wherein a (\\ displaystyle a) negative, and others are positive, then b 3 \u003d c 3 + | a | 3 (\\ displaystyle b ^ (3) \u003d c ^ (3) + | a | ^ (3)), and we get natural solutions c, | a | , b. (\\ displaystyle c, | a |, b.) Therefore, both formulations are equivalent.

Fermat's theorem is generalized by the refuted Euler conjecture and the open Lander - Parkin - Selfridge conjecture.

History

Al-Khojandi tried to prove this theorem for the case in the 10th century, but his proof has not survived.

IN general view the theorem was formulated by Pierre Fermat in 1637 in the margin of Diophantus' Arithmetic. The fact is that Fermat made his own notes in the margins of the mathematical treatises he read and in the same place formulated problems and theorems that came to mind. He wrote down the theorem in question, with the note that the ingenious proof of this theorem he found is too long to be placed in the margins of the book:

On the contrary, it is impossible to decompose a cube into two cubes, a biquadrat into two biquadrats, and in general no degree greater than a square into two degrees with the same exponent. I have found some truly wonderful proof of this, but the margins of the book are too narrow for it.

Original text (lat.)
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duas eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Fermat gives only a proof, as a solution to the problem that reduces to the fourth degree of the theorem n \u003d 4 (\\ displaystyle n \u003d 4), in the 45th commentary on the "Arithmetic" of Diophantus and in a letter to Karkavi (August 1659). In addition, Fermat included the case n \u003d 3 (\\ displaystyle n \u003d 3) to the list of problems solved by the infinite descent method.

Many eminent mathematicians and many amateur amateurs have worked on a complete proof of the Great Theorem; the theorem is considered to be in first place in terms of the number of incorrect "proofs". Nevertheless, these efforts have led to many important results in modern number theory. David Hilbert in his talk "Mathematical Problems" at the II International Congress of Mathematicians (1900) noted that the search for a proof for this seemingly insignificant theorem led to profound results in number theory. In 1908, the German mathematician Wolfskel bequeathed 100,000 German marks to the prover of Fermat's theorem. However, after the First World War, the prize was depreciated.

In the 1980s appeared new approach to solve the problem. From the Mordell conjecture, proved by Faltings in 1983, it follows that the equation a n + b n \u003d c n (\\ displaystyle a ^ (n) + b ^ (n) \u003d c ^ (n)) at n\u003e 3 (\\ displaystyle n\u003e 3) can have only a finite number of relatively simple solutions.

German mathematician Gerhard Fry suggested that Fermat's Last Theorem is a consequence of the Taniyama - Shimura conjecture. This assumption has been proven Ken Ribet .

The last important step in the proof of the theorem was taken by Wiles in September 1994. His 130-page proof was published in the Annals of Mathematics.

Wiles published the first version of his proof in 1993 (after seven years of work), but in it a serious [ which one?] a gap that, with the help of Richard Lawrence Taylor, was quickly eliminated. In 1995, the final version was published. In 2016, Andrew Wiles received the Abel Prize for proving Fermat's Last Theorem.

Colin McLarty noted that perhaps Wiles's proof could be simplified so as not to assume the existence of so-called "big cardinals."

Fermat's theorem also trivially follows from the abc-conjecture, the proof of which was stated by the Japanese mathematician Shinichi Mochizuki; its proof is extremely difficult. There is currently no clear consensus in the mathematical community regarding his work.

Some variations and generalizations

2682440 4 + 15365639 4 + 18796760 4 \u003d 20615673 4. (\\ displaystyle 2682440 ^ (4) + 15365639 ^ (4) + 18796760 ^ (4) \u003d 20615673 ^ (4).)

Later, other solutions were found; the simplest of them:

95800 4 + 217519 4 + 414560 4 \u003d 422481 4. (\\ displaystyle 95800 ^ (4) + 217519 ^ (4) + 414560 ^ (4) \u003d 422481 ^ (4).)

Another popular generalization of Fermat's theorem is the Beale conjecture, formulated in 1993 by an American amateur mathematician who promised $ 1 million for its proof or refutation.

"Fermatists"

The simplicity of the formulation of Fermat's theorem (understandable even for a schoolchild), as well as the complexity of the only known proof (or ignorance of its existence), inspire many to try to find another, simpler proof. People trying to prove Fermat's theorem using elementary methods are called “ fermatists"Or" fermatics ". Fermatists are often not professionals and make mistakes in arithmetic or logical inference, although some present very sophisticated "proofs" in which it is difficult to find an error.

It was so popular to prove Fermat's theorem among amateurs of mathematics that in 1972 the journal Kvant, publishing an article about Fermat's theorem, accompanied it with the following postscript: “The editorial staff of Kvant, The farm will not be considered (and returned). "

The German mathematician Edmund Landau was very bothered by the "fermatists". In order not to be distracted from his main work, he ordered several hundred forms with template text informing that there is an error on a certain line on a certain page, while he instructed his graduate students to find the error and fill in the blanks in the form.

It is noteworthy that individual fermatists are pushing for the publication of their (incorrect) "evidence" in the unscientific press, which inflates its significance into a scientific sensation. However, sometimes such publications appear in respected scientific publications, as a rule, with subsequent refutations. Other examples include:

Fermat's theorem in culture and art

Fermat's Last Theorem has become a symbol of the most difficult scientific problem and as such is often referred to in fiction. The following are some works in which the theorem is not only mentioned, but is an essential part of the plot or ideology of the work.

In the story of Arthur Porges Simon Flagg and the Devil Professor Simon Flegg turns to the devil for a proof of the theorem. A popular science fiction film was made based on this story "The mathematician and the devil" (USSR, production of Tsentrnauchfilm, creative association "Rainbow", director Reitburgh).
- Screen adaptation: short film "Mathematician and the devil" (1972).
AP Kazantsev in his novel Sharper Swords in 1983 offered an original version of the lack of proof of Pierre Fermat himself.
In the television series Star Trek, spaceship captain Jean-Luc Picard was puzzled by Fermat's Last Theorem in the second half of the 24th century. Thus, the filmmakers assumed that Fermat's Last Theorem would not have a solution in the next 400 years. The Royal series with this episode was filmed in 1989 when Andrew Wiles was at the very beginning of his work. In fact, a solution was found just five years later.
In the 1995 Halloween episode of The Simpsons, 2D Homer Simpson accidentally enters the third dimension. During his journey through this strange world, geometric bodies and mathematical formulas, including false equality, float in the air 1782 12 + 1841 12 \u003d 1922 12 (\\ displaystyle 1782 ^ (12) + 1841 ^ (12) \u003d 1922 ^ (12))... A calculator with an accuracy of no more than 10 significant digits confirms this equality: 1782 12 + 1841 12 \u003d 2 541 210 258 614 589 176 288 669 958 142 428 526 657 ≈ 2.541 210 259 ⋅ 10 39, 1922 12 \u003d 2 541 210 259 314 801 410 819 278 649 643 651 567 616 ≈ 2.541 210 259 ⋅ 10 39. (\\ displaystyle (\\ begin (array) (cl) 1782 ^ (12) + 1841 ^ (12) & \u003d 2 \\, 541 \\, 210 \\, 258 \\, 614 \\, 589 \\, 176 \\, 288 \\, 669 \\, 958 \\, 142 \\, 428 \\, 526 \\, 657 \\ approx 2 (,) 541 \\, 210 \\, 259 \\ cdot 10 ^ (39), \\\\ 1922 ^ (12) & \u003d 2 \\, 541 \\ 10 ^ (39). \\ End (array)))

However, even without calculating the exact values, it is easy to see that the equality is not true: the left side is an odd number, and right part - even.

In the first edition of The Art of Computer Programming by Donald Knuth, Fermat's theorem is given as a mathematical exercise at the very beginning of the book and is estimated with a maximum number (50) points, as “A research problem which (as far as the author knew at the time of writing) has not yet been satisfactorily resolved. If the reader finds a solution to this problem, he is urged to publish it; in addition, the author of this book would be very grateful if he was informed of the decision as soon as possible (provided that it is correct). " In the third edition of the book, this exercise already requires knowledge of higher mathematics and is estimated at only 45 points.
In Stieg Larsson's book, The Girl Who Played with Fire, the main character Lisbeth Salander, with rare analytical skills and photographic memory, is engaged as a hobby in proving Fermat's Last Theorem, which she stumbled upon while reading the fundamental work Measurements in Mathematics, in which is also proved by Andrew Wiles. Lisbeth does not want to study a ready-made proof, and his main interest is finding his own solution. Therefore, she devotes all her free time to an independent search for a "wonderful proof" of the theorem of the great Frenchman, but over and over again comes to a dead end. At the end of the book, Lisbeth finds a proof that is not only completely different from that proposed by Wiles, but also so simple that Fermat himself could find it. However, after being shot in the head, she forgets him, and Larsson does not provide any details of this evidence.
The musical "Last Tango Farm", published, created in 2000 by Joshua Rosenblum (eng. Joshua rosenblum) and Joan Lessner based on real story Andrew Wiles. The main character named Daniel Keane completes the proof of the theorem, and the spirit of Fermat himself tries to prevent him.
A few days before his death, Arthur Clarke managed to review the manuscript of the novel The Last Theorem, on which he co-authored with Frederick Paul. The book came out after Clark's death.

Notes

Fermat's theorem // Mathematical encyclopedia (in 5 volumes). - M.: Soviet Encyclopedia, 1985 .-- T. 5.
Diophantus of Alexandria. Arithmeticorum libri sex, et de numeris multangulis liber unus. Cum commentariis C.G. Bacheti V.C. & observationibus D.P. de Fermat senatoris Tolosani. Toulouse, 1670, pp. 338-339.
Fermat a Carcavi. Aout 1659. Oeuvres de Fermat. Tome II. Paris: Tannery & Henry, 1904, pp. 431-436.
Yu Yu Machis. On the supposed proof of Euler // Mathematical notes. - 2007. - T. 82, No. 3. - S. 395-400. English translation: J. J. Mačys. On Euler's hypothetical proof (English) // Mathematical Notes: journal. - 2007. - Vol. 82, no. 3-4. - P. 352-356. - DOI: 10.1134 / S0001434607090088.
David Gilbert. Mathematical problems:

The problem of proving this undecidability is a striking example of what a special and seemingly insignificant problem can have a stimulating effect on science. For, prompted by Fermat's problem, Kummer came to the introduction of ideal numbers and to the discovery of a theorem on the unique decomposition of numbers in circular fields into ideal prime factors - a theorem that is now, thanks to generalizations to any algebraic number domain obtained by Dedekind and Kronecker, is central to modern theory numbers and the value of which goes far beyond number theory into the field of algebra and function theory.
Yu.P. Soloviev Taniyama's hypothesis and Fermat's last theorem // Soros Educational Journal. - ISSEP, 1998. - T. 4, No. 2. - S. 135-138.
Wiles, Andrew. Modular elliptic curves and Fermat’s last theorem (English) // Annals of Mathematics: journal. - 1995. - Vol. 141, no. 3. - P. 443-551. (eng.)

Envious people claim that the French mathematician Pierre Fermat wrote his name in history with just one phrase. In the margins of the manuscript formulating the famous theorem in 1637, he made a note: "I have found an amazing solution, but there is not enough space here to put it." Then an amazing mathematical race began, in which, along with outstanding scientists, an army of amateurs joined.

What is the insidiousness of Fermat's problem? At first glance, even a schoolchild can understand it.

It is based on the well-known Pythagorean theorem: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: x 2 + y 2 \u003d z 2. Fermat argued: the equation for any powers greater than two has no solution in integers.

It would seem simple. Reach out your hand and here's the answer. No wonder academies different countries, scientific institutions, even newspaper offices were inundated with tens of thousands of proofs. Their number is unprecedented, second only to the projects of "perpetual motion machines". But if serious science has not considered these crazy ideas for a long time, then the work of "farmers" is honestly and interestedly studying. And, alas, he finds mistakes. They say that in more than three centuries a whole mathematical cemetery of solutions to the theorem has formed.

No wonder they say: the elbow is close, and you won't bite. Years, decades, centuries passed, and Fermat's task seemed more and more surprising and tempting. Seemingly unpretentious, it turned out to be too tough for the rapidly building muscle progress. The man had already split the atom, got to the gene, set foot on the moon, but Fermat was not given, continuing to attract descendants with false hopes.

However, attempts to overcome the scientific peak were not in vain. The first step was taken by the great Euler, proving the theorem for the fourth degree, then for the third. At the end of the 19th century, the German Ernst Kummer brought the number of degrees to one hundred. Finally, armed with computers, scientists increased that figure to 100,000. But Fermat was talking about any degrees. This was the whole problem.

Of course, scientists were tormented by the task not because of sports interest. The famous mathematician David Hilbert said that a theorem is an example of how a seemingly insignificant problem can have a huge impact on science. Working on it, scientists opened completely new mathematical horizons, for example, the foundations of number theory, algebra, and function theory were laid.

And yet the Great Theorem was subdued in 1995. Its solution was presented by an American from Princeton University, Andrew Wiles, and it is officially recognized by the scientific community. He gave more than seven years of his life to find proof. According to scientists, this outstanding work brought together the works of many mathematicians, restoring the lost connections between its various sections.

So, the summit has been taken, and science has received the answer, - the scientific secretary of the Department of Mathematics of the Russian Academy of Sciences, Doctor of Technical Sciences Yuri Vishnyakov told the RG correspondent. - The theorem is proved, albeit not in the simplest way, on which Fermat himself insisted. And now those who wish can print their versions.

However, the family of "farmers" is not at all going to accept Wiles's proof. No, they do not refute the American's decision, because it is very complex, and therefore understandable only to a narrow circle of specialists. But not a week goes by without a new revelation of another enthusiast appearing on the Internet, "who has finally put an end to the long-term epic."

By the way, just yesterday, one of the oldest "farmers" in our country, Vsevolod Yarosh, phoned the editorial office of RG: mathematician Academician Arnold with a request to publish about this in a scientific journal. Now I am waiting for an answer. I am in correspondence with the French Academy of Sciences about this. "

And just now, as reported in a number of media outlets, with "light grace has revealed the great mystery of mathematics", another enthusiast - the former general designer of PO "Polet" from Omsk, Doctor of Technical Sciences Alexander Ilyin. The solution turned out to be so simple and short that it fit into a small section of the newspaper space of one of the central publications.

The editorial board of "RG" applied to the leading in the country Institute of Mathematics. Steklov Russian Academy of Sciences with a request to evaluate this solution. Scientists were categorical: you cannot comment on a newspaper publication. But after much persuasion and taking into account the increased interest in the famous problem, they agreed. According to them, several fundamental mistakes were made in the next published proof. By the way, even a student of the Faculty of Mathematics could have noticed them.

And yet the editors wanted to get first-hand information. Moreover, yesterday Ilyin had to present his proof at the Aviation and Aeronautics Academy. However, it turned out that few people know about such an academy, even among specialists. And when, nevertheless, with the greatest difficulty, he managed to find the phone number of the scientific secretary of this organization, then, as it turned out, he did not even suspect that it was with them that such a historic event should take place. In a word, the "RG" correspondent did not manage to witness the world sensation.

Pierre Fermat argued that:

it is impossible to decompose a cube into two cubes or a biquadrat into two biquadrats, and in general it is impossible to decompose any degree greater than two into two degrees with the same exponent.

How do we approach the proof of this statement by Fermat?

(picture to attract attention)

Let's imagine that we have found or built a right-angled triangle with the following sides: legs -, and a hypotenuse where (p, q, k, n) - natural numbers. Then, by the Pythagorean theorem, we get or. Thus, if we find or construct such a triangle, we will refute Fermat. If we prove that such a triangle does not exist, then we will prove the theorem.

Since the statement deals with natural numbers, we will find what the difference of the squares of two odd natural numbers is equal to. Those. solve the equation. To do this, we construct right-angled triangles, the hypotenuse of which is equal, and the leg is equal, where and (a\u003e b)... Then, by the Pythagorean theorem, you can calculate the second leg by the formula (1) , or (2) ... We got that the sides of these triangles are equal and. So we can iterate over all pairs of numbers a and b from a natural set (we will call these numbers “generators” of this identity) and obtain all possible triangles with given properties,. Let us prove the necessity of this solution. Let's rewrite (1) as . Since Z and Y are odd numbers, so you can write (Z - Y) \u003d 2b and (Z + Y) \u003d 2a. Solving them for Z and Y, we get Z \u003d (a + b) and Y \u003d (a - b). Then we can write that X \u003d 4ab and substituting these values \u200b\u200binto (1) , we get.

Note

To avoid getting similar triangles, and given that Z and Y - odd numbers by condition, numbers a and b must be coprime and of different parity. In what follows, we will assume that even is the number a... To arrange the distribution of right-angled triangles in the set of natural numbers N, we will proceed as follows: from this set we subtract all numbers that are even powers of natural numbers. We denote this set, where n - natural number. Then, from the remaining natural numbers, subtract all numbers that are odd (≥3) powers of natural numbers and denote the set of these numbers as. The remaining natural numbers will form a set, the numbers of which are natural numbers in the first degree. Let's denote this set. Obviously, the connection of these 3 sets is a set of natural numbers, or. Let us represent the set as a series \u003d (1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, ………). We represent the sets and in the form of series. Then the set will be a matrix consisting of an infinite number of rows, each row will consist of numbers in the row raised to the power 2n, and n - there is a line number. So the first line consists of the squares of all numbers in the row, the second line consists of 4 powers of these numbers, etc. Consider a set that will be a matrix consisting of an infinite number of rows, each row of which will consist of numbers in the series raised to the power 2n + 1... (n is the line number). So the first row of this matrix consists of cubes of numbers of the row, the second row consists of numbers of the row in the fifth power, etc. Consider the set. Because , then we will use the same algorithm for constructing triangles (see above). Let's find the "generators" of the identity, These will be numbers, where, we will compose the identity: (3) , we got a lot of right-angled triangles with integer sides. Here is the hypotenuse, is the leg and is the second leg. To refute Fermat's statement, it is necessary that the parties X, Y, Z of the desired triangle were (4) ... Where (p, q, k, n) are natural numbers. By the Pythagorean theorem, we will have or

and Fermat's claim will be refuted. The identity shows that. Consider the last equality, in this equality « p"For any values" a and b"Is not a natural number if. This means that in the considered set of triangles there is not a single triangle with the required sides (4) .
Now consider the set. We denote (2n + 1) as " m», Then in the set we obtain right-angled triangles described by the identity (6) ... If we can build a right triangle X, Y, Z with the parties (7) , where, then we refute Fermat's statement, since by the Pythagorean theorem and (p, q and k) are natural numbers. It is necessary that. Considering the last equality, we note that “ p"Cannot be a natural number for any value" a and b", , if a . This means that in this set of triangles there is not a single triangle with the required sides (7) .

However, it can be seen from the above that the whole proof is reduced to the analysis of the number, where "" for any natural " a and b"Is not a natural number to the degree" m / 2". Or (8) under the same conditions will not be a natural number to the power of "m". The proof shows that the "generators" of the identity (6) are the numbers "" from the series But, analyzing (8) , you can substitute a number instead of "". Since there is an even number, (see Note), then it is a natural number. After substituting it in (8) we get, that is, natural numbers to the power "m". Making the above substitution into the identity (6) , and, denoting by, we obtain the following identity:. We got a lot of right-angled triangles with sides. If (k, q, p) are natural numbers in an odd degree, i.e. where r is any odd number, and. To refute Fermat it is necessary that: In the last equality for any natural a and b, are natural numbers, but the first two equalities are impossible, since if “ m and r»Any odd numbers, then are irrational numbers, and the numbers in brackets are natural numbers. If (k, q, p) are natural numbers in an even power, i.e. , then we get the following equalities (5) ... In this variant, the last equality is impossible, since extracting the root m of degree from both sides of the equality, we obtain, i.e. in brackets is an irrational number, and a natural number. This means that the "required" triangle was not found in this set either. This means that for any odd « m"Fermat's statement is true, which means it is true for all simple exponents" m ≥ 3 ".

It remains to find a proof of the theorem for even exponents. Of (5) it follows that if in the canonical decomposition of an even exponent there is an odd prime number, then Fermat's statement is true for this degree. Obviously, this condition is met by all even numbers, except for the number 4 "And multiples of four, ie. 8, 16, 32, 64 … etc. There is only a prime number in the expansion of these numbers 2 ... Therefore, the above evidence does not provide an answer for these degrees.

Hence, it remains to prove the theorem for n \u003d 4". It can be assumed that Fermat had general evidence, but not complete. Maybe that's why he didn't write down his proof. And only a few years later, having created his method of "infinite or indefinite descent", he proved that there is no right-angled triangle with integer sides, whose area would be equal to the square of a natural number. After this, the proof of the theorem for n \u003d 4"Was not difficult. Fermat recorded this proof. And the theorem was proved completely.

Tags: Fermat's theorem, short proof