Why can't they prove Fermat's theorem? Let's expose! Has Fermat's Last Theorem been proven? Work by Shimura and Taniyama

Since few people have mathematical thinking, I will talk about the largest scientific discovery - the elementary proof of Fermat's Last Theorem - in the most understandable, school language.

The proof was found for a special case (for a simple degree n>2), to which (and to the case n=4) all cases with composite n can easily be reduced.

So, we need to prove that the equation A^n=C^n-B^n has no solution in integers. (Here the ^ sign means degree.)

The proof is carried out in a number system with a simple base n. In this case, the last digits in each multiplication table are not repeated. In the usual decimal system, the situation is different. For example, when multiplying the number 2 by both 1 and 6, both products - 2 and 12 - end in the same digits (2). And, for example, in the septenary system for the number 2, all the last digits are different: 0x2=...0, 1x2=...2, 2x2=...4, 3x2=...6, 4x2=...1, 5x2=...3, 6x2=...5, with a set of last digits 0, 2, 4, 6, 1, 3, 5.

Thanks to this property, for any number A that does not end in zero (and in Fermat’s equality, the last digit of the numbers A, or B, after dividing the equality by the common divisor of the numbers A, B, C is not equal to zero), it is possible to select a factor g such that the number Ag will have an arbitrarily long ending of the form 000...001. It is by this number g that we multiply all the base numbers A, B, C in Fermat’s equality. In this case, we will make the unit ending quite long, namely two digits longer than the number (k) of zeros at the end of the number U=A+B-C.

The number U is not equal to zero - otherwise C=A+B and A^n<(А+В)^n-B^n, т.е. равенство Ферма является неравенством.

This, in fact, is all the preparation of Fermat’s equality for a brief and final study. The only thing we will do is rewrite the right-hand side of Fermat’s equality – C^n-B^n – using the school decomposition formula: C^n-B^n=(C-B)P, or aP. And since further we will operate (multiply and add) only with the digits of the (k+2)-digit endings of the numbers A, B, C, then we can not take their leading parts into account and simply discard them (leaving only one fact in memory: the left side of Fermat's equality is a POWER).

The only thing worth mentioning is the last digits of the numbers a and P. In Fermat’s original equality, the number P ends with the number 1. This follows from the formula of Fermat’s little theorem, which can be found in reference books. And after multiplying Fermat’s equality by the number g^n, the number P is multiplied by the number g to the power n-1, which, according to Fermat’s little theorem, also ends in the number 1. So in the new equivalent Fermat equality, the number P ends in 1. And if A ends in 1, then A^n also ends in 1 and, therefore, the number a also ends in 1.

So, we have a starting situation: the last digits A, a, P of the numbers A, a, P end in the number 1.

Well, then begins a cute and fascinating operation, called in preference a “mill”: by introducing into consideration the subsequent numbers a"", a""" and so on, numbers a, we extremely “easily” calculate that they are all also equal to zero! Word I put “easy” in quotes, because humanity could not find the key to this “easy” for 350 years! ^(k+2). It’s not worth paying attention to the second term in this sum - after all, in the further proof we discarded all the digits after the (k+2)th in the numbers (and this radically simplifies the analysis)! So after discarding the head parts numbers, Fermat’s equality takes the form: ...1=aq^(n-1), where a and q are not numbers, but just the endings of the numbers a and q! (I don’t introduce new notations, as this makes it difficult to read.)

The last philosophical question remains: why can the number P be represented as P=q^(n-1)+Qn^(k+2)? The answer is simple: because any integer P with 1 at the end can be represented in this form, and IDENTICALLY. (It can be represented in many other ways, but we don’t need that.) Indeed, for P=1 the answer is obvious: P=1^(n-1). For Р=hn+1, the number q=(n-h)n+1, which is easy to verify by solving the equation [(n-h)n+1]^(n-1)==hn+1 using two-digit endings. And so on (but we don’t need further calculations, since we only need to represent numbers of the form P=1+Qn^t).

Phew! Well, the philosophy is over, you can move on to calculations at the second-grade level, perhaps just remember Newton’s binomial formula once again.

So, let’s introduce the number a"" (in the number a=a""n+1) and use it to calculate the number q"" (in the number q=q""n+1):
...01=(a""n+1)(q""n+1)^(n-1), or...01=(a""n+1)[(n-q"")n+ 1], whence q""=a"".

And now the right side of Fermat’s equality can be rewritten as:
A^n=(a""n+1)^n+Dn^(k+2), where the value of the number D does not interest us.

Now we come to the decisive conclusion. The number a""n+1 is the two-digit ending of the number A and, THEREFORE, according to a simple lemma, UNIQUELY determines the THIRD digit of the degree A^n. And moreover, from the expansion of Newton's binomial
(a""n+1)^n, taking into account that to each term of the expansion (except for the first, which cannot change the weather!) a SIMPLE factor n (the number base!) is added, it is clear that this third digit is equal to a"" . But by multiplying Fermat’s equality by g^n, we turned k+1 digits before the last 1 in number A into 0. And, therefore, a""=0!!!

Thus, we completed the cycle: having entered a"", we found that q""=a"", and finally a""=0!

Well, it remains to say that after carrying out completely similar calculations and the next k digits, we obtain the final equality: the (k + 2)-digit ending of the number a, or C-B, just like the number A, is equal to 1. But then the (k+2)th digit of the number C-A-B is EQUAL to zero, while it is NOT EQUAL to zero!!!

That, in fact, is all the proof. To understand it, it is not at all necessary to have a higher education and, especially, to be a professional mathematician. However, the professionals remain silent...

Readable text of the full proof is located here:

Reviews

Hello, Victor. I liked your resume. “Don’t let die before death” sounds great, of course. To be honest, I was stunned by my encounter with Fermat’s theorem in Prose! Does she belong here? There are scientific, popular science and teapot sites. Otherwise, thank you for your literary work.
Best regards, Anya.

Dear Anya, despite the rather strict censorship, Prose allows you to write ABOUT EVERYTHING. The situation with Fermat’s theorem is as follows: large mathematical forums treat Fermatists askance, with rudeness, and in general treat them as best they can. However, I presented the latest version of the proof in small Russian, English and French forums. No one has put forward any counter-arguments yet, and, I’m sure, no one will put forward any (the evidence has been checked very carefully). On Saturday I will publish a philosophical note about the theorem.
There are almost no boors in prose, and if you don’t hang around with them, then pretty soon they will fall off.
Almost all of my works are presented on Prose, so I also included the proof here.
See you later,

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

Yes, this theorem is very well known and, in a sense, has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this happened back in 1995. But first things first.

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat’s Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat’s last theorem was practically over.

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

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

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

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

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

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

Dear. . . . . . . .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

source

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

SCIENCE AND TECHNOLOGY NEWS

UDC 51:37;517.958

A.V. Konovko, Ph.D.

Academy of State Fire Service of the Ministry of Emergency Situations of Russia FERMA'S GREAT THEOREM HAS BEEN PROVEN. OR NOT?

For several centuries, it was not possible to prove that the equation xn+yn=zn for n>2 is unsolvable in rational numbers, and therefore in integers. This problem was born under the authorship of the French lawyer Pierre Fermat, who at the same time was professionally engaged in mathematics. Her decision is credited to the American mathematics teacher Andrew Wiles. This recognition lasted from 1993 to 1995.

THE GREAT FERMA"S THEOREM IS PROVED. OR NO?

The dramatic history of Fermat"s last theorem proving is considered. It took almost four hundred years. Pierre Fermat wrote little. He wrote in compressed style. Besides he did not publish his researches. The statement that equation xn+yn=zn is unsolvable on sets of rational numbers and integers if n>2 was attended by Fermat"s commentary that he has found indeed remarkable proving to this statement. The descendants were not reached by this proving. Later this statement was called Fermat "s last theorem. The world best mathematicians broke lance over this theorem without result. In the seventies the French mathematician member of Paris Academy of Sciences Andre Veil laid down new approaches to the solution. In 23 of June, in 1993, at theory of numbers conference in Cambridge, the mathematician of Princeton University Andrew Whiles announced that the Fermat"s last theorem proving is completed. However it was early to triumph.

In 1621, the French writer and lover of mathematics Claude Gaspard Bachet de Meziriak published the Greek treatise "Arithmetic" of Diophantus with a Latin translation and commentary. The luxurious “Arithmetic,” with unusually wide margins, fell into the hands of twenty-year-old Fermat and became his reference book for many years. In its margins he left 48 notes containing the facts he discovered about the properties of numbers. Here, in the margins of “Arithmetic,” Fermat’s great theorem was formulated: “It is impossible to decompose a cube into two cubes or a biquadrate into two biquadrates, or in general a power greater than two into two powers with the same exponent; I found a truly wonderful proof of this, which due to lack of space cannot fit in these fields." By the way, in Latin it looks like this: “Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”

The great French mathematician Pierre Fermat (1601-1665) developed a method for determining areas and volumes and created a new method of tangents and extrema. Along with Descartes, he became the creator of analytical geometry, together with Pascal stood at the origins of the theory of probability, in the field of the infinitesimal method he gave the general rule of differentiation and proved in general form the rule of integration of a power function... But, most importantly, one of the most important mysterious and dramatic stories that have ever shocked mathematics - the story of the proof of Fermat's last theorem. Now this theorem is expressed in the form of a simple statement: the equation xn + yn = zn for n>2 is unsolvable in rational numbers, and therefore in integers. By the way, for the case n = 3, the Central Asian mathematician Al-Khojandi tried to prove this theorem in the 10th century, but his proof did not survive.

A native of the south of France, Pierre Fermat received a legal education and from 1631 served as an adviser to the parliament of the city of Toulouse (i.e., the highest court). After a working day within the walls of parliament, he took up mathematics and immediately plunged into a completely different world. Money, prestige, public recognition - none of this mattered to him. Science never became a livelihood for him, did not turn into a craft, always remaining just an exciting game of the mind, understandable only to a few. He carried on his correspondence with them.

Fermat never wrote scientific papers in our usual sense. And in his correspondence with friends there is always some challenge, even a kind of provocation, and by no means an academic presentation of the problem and its solution. That’s why many of his letters subsequently came to be called a challenge.

Perhaps this is precisely why he never realized his intention to write a special essay on number theory. Meanwhile, this was his favorite area of ​​mathematics. It was to her that Fermat dedicated the most inspired lines of his letters. “Arithmetic,” he wrote, “has its own field, the theory of integers. This theory was only slightly touched upon by Euclid and was not sufficiently developed by his followers (unless it was contained in those works of Diophantus, which the ravages of time have deprived us of). Arithmeticians, therefore, must develop and renew it."

Why was Fermat himself not afraid of the destructive effects of time? He wrote little and always very concisely. But, most importantly, he did not publish his work. During his lifetime they circulated only in manuscripts. It is not surprising, therefore, that Fermat’s results on number theory have reached us in scattered form. But Bulgakov was probably right: great manuscripts don’t burn! Fermat's work remains. They remained in his letters to friends: the Lyon mathematics teacher Jacques de Billy, the mint employee Bernard Freniquel de Bessy, Marcenny, Descartes, Blaise Pascal... What remained was Diophantus' "Arithmetic" with his comments in the margins, which after Fermat's death were included together with comments by Bachet into the new edition of Diophantus, published by his eldest son Samuel in 1670. Only the evidence itself has not survived.

Two years before his death, Fermat sent his friend Carcavi a letter of testament, which went down in the history of mathematics under the title “Summary of new results in the science of numbers.” In this letter, Fermat proved his famous statement for the case n = 4. But then he was most likely not interested in the statement itself, but in the method of proof he discovered, which Fermat himself called infinite or indefinite descent.

Manuscripts don't burn. But, if not for the dedication of Samuel, who after his father’s death collected all his mathematical sketches and small treatises, and then published them in 1679 under the title “Miscellaneous Mathematical Works,” learned mathematicians would have had to discover and rediscover a lot. But even after their publication, the problems posed by the great mathematician lay motionless for more than seventy years. And this is not surprising. In the form in which they appeared in print, the number-theoretic results of P. Fermat appeared before specialists in the form of serious problems that were not always clear to contemporaries, almost without proof, and indications of internal logical connections between them. Perhaps, in the absence of a coherent, well-thought-out theory lies the answer to the question why Fermat himself never decided to publish a book on number theory. Seventy years later, L. Euler became interested in these works, and this was truly their second birth...

Mathematics paid dearly for Fermat's peculiar manner of presenting his results, as if deliberately omitting their proofs. But, if Fermat claimed that he had proved this or that theorem, then this theorem was subsequently proven. However, there was a hitch with the great theorem.

A mystery always excites the imagination. Entire continents were conquered by the mysterious smile of Gioconda; The theory of relativity, as the key to the mystery of space-time connections, has become the most popular physical theory of the century. And we can safely say that there was no other mathematical problem that was as popular as it was ___93

Scientific and educational problems of civil protection

What is Fermat's theorem? Attempts to prove it led to the creation of an extensive branch of mathematics - the theory of algebraic numbers, but (alas!) the theorem itself remained unproven. In 1908, the German mathematician Wolfskehl bequeathed 100,000 marks to anyone who could prove Fermat's theorem. This was a huge amount for those times! In one moment you could become not only famous, but also get fabulously rich! It is not surprising, therefore, that high school students even in Russia, far from Germany, vying with each other, rushed to prove the great theorem. What can we say about professional mathematicians! But...in vain! After the First World War, money became worthless, and the flow of letters with pseudo-evidence began to dry up, although, of course, it never stopped. They say that the famous German mathematician Edmund Landau prepared printed forms to send to authors of proofs of Fermat’s theorem: “There is an error on page ..., in line ....” (The assistant professor was tasked with finding the error.) There were so many oddities and anecdotes related to the proof of this theorem that one could compile a book out of them. The latest anecdote is A. Marinina’s detective story “Coincidence of Circumstances,” filmed and shown on the country’s television screens in January 2000. In it, our compatriot proves a theorem unproved by all his great predecessors and claims a Nobel Prize for it. As you know, the inventor of dynamite ignored mathematicians in his will, so the author of the proof could only claim the Fields Gold Medal, the highest international award approved by mathematicians themselves in 1936.

In the classic work of the outstanding Russian mathematician A.Ya. Khinchin, dedicated to Fermat’s great theorem, provides information on the history of this problem and pays attention to the method that Fermat could have used to prove his theorem. A proof for the case n = 4 and a brief review of other important results are given.

But by the time the detective story was written, and even more so by the time it was filmed, the general proof of the theorem had already been found. On June 23, 1993, at a conference on number theory in Cambridge, Princeton mathematician Andrew Wiles announced that Fermat's Last Theorem had been proven. But not at all as Fermat himself “promised”. The path that Andrew Wiles took was not based on the methods of elementary mathematics. He studied the so-called theory of elliptic curves.

To get an idea of ​​elliptic curves, you need to consider a plane curve defined by a third-degree equation

Y(x,y) = a30X + a21x2y+ ... + a1x+ a2y + a0 = 0. (1)

All such curves are divided into two classes. The first class includes those curves that have sharpening points (such as the semi-cubic parabola y2 = a2-X with the sharpening point (0; 0)), self-intersection points (like the Cartesian sheet x3+y3-3axy = 0, at the point (0; 0)), as well as curves for which the polynomial Dx,y) is represented in the form

f(x^y)=:fl(x^y)■:f2(x,y),

where ^(x,y) and ^(x,y) are polynomials of lower degrees. Curves of this class are called degenerate curves of the third degree. The second class of curves is formed by non-degenerate curves; we will call them elliptic. These may include, for example, the Agnesi Curl (x2 + a2)y - a3 = 0). If the coefficients of the polynomial (1) are rational numbers, then the elliptic curve can be transformed to the so-called canonical form

y2= x3 + ax + b. (2)

In 1955, the Japanese mathematician Y. Taniyama (1927-1958), within the framework of the theory of elliptic curves, managed to formulate a hypothesis that opened the way for the proof of Fermat’s theorem. But neither Taniyama himself nor his colleagues suspected this at the time. For almost twenty years this hypothesis did not attract serious attention and became popular only in the mid-70s. According to the Taniyama conjecture, every elliptic

a curve with rational coefficients is modular. However, so far the formulation of the hypothesis tells little to the meticulous reader. Therefore, some definitions are required.

Each elliptic curve can be associated with an important numerical characteristic - its discriminant. For a curve given in the canonical form (2), the discriminant A is determined by the formula

A = -(4a + 27b2).

Let E be some elliptic curve given by equation (2), where a and b are integers.

For a prime number p, consider the comparison

y2 = x3 + ax + b(mod p), (3)

where a and b are the remainders from dividing the integers a and b by p, and let us denote by np the number of solutions to this comparison. The numbers pr are very useful in studying the question of the solvability of equations of the form (2) in integers: if some pr is equal to zero, then equation (2) has no integer solutions. However, it is possible to calculate numbers only in the rarest cases. (At the same time it is known that р-п|< 2Vp (теоремаХассе)).

Let us consider those prime numbers p that divide the discriminant A of the elliptic curve (2). It can be proven that for such p the polynomial x3 + ax + b can be written in one of two ways:

x3 + ax + b = (x + a)2 (x + ß)(mod P)

x3 + ax + b = (x + y)3 (mod p),

where a, ß, y are some remainders from division by p. If for all primes p dividing the discriminant of the curve, the first of the two indicated possibilities is realized, then the elliptic curve is called semistable.

The prime numbers dividing the discriminant can be combined into what is called an elliptic curve jig. If E is a semistable curve, then its conductor N is given by the formula

where for all prime numbers p > 5 dividing A, the exponent eP is equal to 1. Exponents 82 and 83 are calculated using a special algorithm.

Essentially, this is all that is necessary to understand the essence of the proof. However, Taniyama’s hypothesis contains a complex and, in our case, key concept of modularity. Therefore, let's forget about elliptic curves for a moment and consider the analytic function f (that is, the function that can be represented by a power series) of the complex argument z, given in the upper half-plane.

We denote by H the upper complex half-plane. Let N be a natural number and k be an integer. A modular parabolic form of weight k of level N is an analytic function f(z) defined in the upper half-plane and satisfying the relation

f = (cz + d)kf (z) (5)

for any integers a, b, c, d such that ae - bc = 1 and c is divisible by N. In addition, it is assumed that

lim f (r + it) = 0,

where r is a rational number, and that

The space of modular parabolic forms of weight k of level N is denoted by Sk(N). It can be shown that it has finite dimension.

In what follows, we will be especially interested in modular parabolic forms of weight 2. For small N, the dimension of the space S2(N) is presented in Table. 1. In particular,

Dimensions of the space S2(N)

Table 1

N<10 11 12 13 14 15 16 17 18 19 20 21 22

0 1 0 0 1 1 0 1 0 1 1 1 2

From condition (5) it follows that % + 1) = for each form f e S2(N). Therefore, f is a periodic function. Such a function can be represented as

Let us call a modular parabolic form A^) in S2(N) proper if its coefficients are integers satisfying the relations:

a g ■ a = a g+1 ■ p ■ c Г_1 for a simple p that does not divide the number N; (8)

(ap) for a prime p dividing the number N;

atn = at an, if (t,n) = 1.

Let us now formulate a definition that plays a key role in the proof of Fermat’s theorem. An elliptic curve with rational coefficients and conductor N is called modular if there is such an eigenform

f (z) = ^anq" g S2(N),

that ap = p - pr for almost all prime numbers p. Here n is the number of comparison solutions (3).

It is difficult to believe in the existence of even one such curve. It is quite difficult to imagine that there would be a function A(r) that satisfies the listed strict restrictions (5) and (8), which would be expanded into series (7), the coefficients of which would be associated with practically incomputable numbers Pr. But Taniyama’s bold hypothesis did not at all cast doubt on the fact of their existence, and the empirical material accumulated over time brilliantly confirmed its validity. After two decades of almost complete oblivion, Taniyama's hypothesis received a kind of second wind in the works of the French mathematician, member of the Paris Academy of Sciences Andre Weil.

Born in 1906, A. Weil eventually became one of the founders of a group of mathematicians who acted under the pseudonym N. Bourbaki. Since 1958, A. Weil became a professor at the Princeton Institute for Advanced Study. And the emergence of his interest in abstract algebraic geometry dates back to this same period. In the seventies he turned to elliptic functions and Taniyama's conjecture. The monograph on elliptic functions was translated here in Russia. He is not alone in his hobby. In 1985, the German mathematician Gerhard Frey proposed that if Fermat's theorem is false, that is, if there is a triple of integers a, b, c such that a" + bn = c" (n > 3), then the elliptic curve

y2 = x (x - a")-(x - cn)

cannot be modular, which contradicts Taniyama's conjecture. Frey himself failed to prove this statement, but soon the proof was obtained by the American mathematician Kenneth Ribet. In other words, Ribet showed that Fermat's theorem is a consequence of Taniyama's conjecture.

He formulated and proved the following theorem:

Theorem 1 (Ribet). Let E be an elliptic curve with rational coefficients and having a discriminant

and conductor

Let us assume that E is modular and let

/ (r) = q + 2 aAn e ^ (N)

is the corresponding proper form of level N. We fix a prime number £, and

р:еР =1;- " 8 р

Then there is such a parabolic form

/(g) = 2 dnqn e N)

with integer coefficients such that the differences an - dn are divisible by I for all 1< п<ад.

It is clear that if this theorem is proven for a certain exponent, then it is thereby proven for all exponents divisible by n. Since every integer n > 2 is divisible either by 4 or by an odd prime number, we can therefore limit ourselves to the case when the exponent is either 4 or an odd prime number. For n = 4, an elementary proof of Fermat's theorem was obtained first by Fermat himself, and then by Euler. Thus, it is enough to study the equation

a1 + b1 = c1, (12)

in which the exponent I is an odd prime number.

Now Fermat's theorem can be obtained by simple calculations (2).

Theorem 2. Fermat's last theorem follows from Taniyama's conjecture for semistable elliptic curves.

Proof. Let's assume that Fermat's theorem is false, and let there be a corresponding counterexample (as above, here I is an odd prime). Let us apply Theorem 1 to the elliptic curve

y2 = x (x - ae) (x - c1).

Simple calculations show that the conductor of this curve is given by the formula

Comparing formulas (11) and (13), we see that N = 2. Therefore, by Theorem 1 there is a parabolic form

lying in space 82(2). But by virtue of relation (6), this space is zero. Therefore, dn = 0 for all n. At the same time, a^ = 1. Therefore, the difference ag - dl = 1 is not divisible by I and we arrive at a contradiction. Thus, the theorem is proven.

This theorem provided the key to the proof of Fermat's Last Theorem. And yet the hypothesis itself remained still unproven.

Having announced on June 23, 1993, the proof of the Taniyama conjecture for semistable elliptic curves, which include curves of the form (8), Andrew Wiles was in a hurry. It was too early for mathematicians to celebrate their victory.

The warm summer quickly ended, the rainy autumn was left behind, and winter came. Wiles wrote and rewrote the final version of his proof, but meticulous colleagues found more and more inaccuracies in his work. And so, in early December 1993, a few days before Wiles' manuscript was to go to press, serious gaps in his evidence were again discovered. And then Wiles realized that he couldn’t fix anything in a day or two. This required serious improvement. The publication of the work had to be postponed. Wiles turned to Taylor for help. “Working on the mistakes” took more than a year. The final version of the proof of the Taniyama conjecture, written by Wiles in collaboration with Taylor, was published only in the summer of 1995.

Unlike the hero A. Marinina, Wiles did not apply for the Nobel Prize, but still... he should have been awarded some kind of award. But which one? Wiles was already in his fifties at that time, and Fields’ gold medals are awarded strictly until the age of forty, when the peak of creative activity has not yet passed. And then they decided to establish a special award for Wiles - the silver badge of the Fields Committee. This badge was presented to him at the next congress on mathematics in Berlin.

Of all the problems that can, with greater or lesser probability, take the place of Fermat's last theorem, the problem of the closest packing of balls has the greatest chance. The problem of the densest packing of balls can be formulated as the problem of how to most economically fold oranges into a pyramid. Young mathematicians inherited this task from Johannes Kepler. The problem arose in 1611, when Kepler wrote a short essay “On Hexagonal Snowflakes.” Kepler's interest in the arrangement and self-organization of particles of matter led him to discuss another issue - the densest packing of particles, in which they occupy the smallest volume. If we assume that the particles have the shape of balls, then it is clear that no matter how they are located in space, there will inevitably remain gaps between them, and the question is to reduce the volume of gaps to a minimum. In the work, for example, it is stated (but not proven) that such a shape is a tetrahedron, the coordinate axes inside which determine the basic orthogonality angle of 109°28", and not 90°. This problem is of great importance for particle physics, crystallography and other branches of natural science .

Literature

1. Weil A. Elliptic functions according to Eisenstein and Kronecker. - M., 1978.

2. Soloviev Yu.P. Taniyama's conjecture and Fermat's last theorem // Soros educational journal. - No. 2. - 1998. - P. 78-95.

3. Singh S. Fermat’s Last Theorem. The story of a mystery that has occupied the world's best minds for 358 years / Trans. from English Yu.A. Danilova. M.: MTsNMO. 2000. - 260 p.

4. Mirmovich E.G., Usacheva T.V. Quaternion algebra and three-dimensional rotations // This journal No. 1(1), 2008. - P. 75-80.