The History of Farm's Great Theorem. Felix Kirsanov

Envious people claim that the French mathematician Pierre Fermat entered his name in history with just one phrase. In the margin of the manuscript with the formulation of the famous theorem in 1637, he made a note: "I found an amazing solution, but there is not enough space 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, it is clear even to a schoolboy.

It is based on the well-known Pythagorean theorem: in a right 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 that an equation with any powers greater than two has no solution in integers.

It would seem simple. Reach out your hand and here is the answer. Not surprisingly, the academies different countries, scientific institutes, even newspaper editorial offices were inundated with tens of thousands of evidence. Their number is unprecedented, second only to projects of "perpetual motion machines". But if serious science has not considered these crazy ideas for a long time, then the works of "fermists" are honestly and interestedly studying. And, alas, finds errors. It is said that for more than three centuries a whole mathematical cemetery of solutions to the theorem has been formed.

No wonder they say: the elbow is close, but you won’t bite. Years, decades, centuries passed, and Fermat's problem seemed more and more surprising and tempting. It seems to be unpretentious, it turned out to be too tough for the progress that is rapidly building up muscles. Man has already split the atom, got to the gene, set foot on the moon, but Fermat did not give in, continuing to beckon his descendants with false hopes.

However, attempts to overcome the scientific pinnacle 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 this figure to 100,000. But Fermat spoke of any degrees. That was the whole point.

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 up 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. Her 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 links between its different sections.

So, the summit has been taken, and science has received an 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 has been proved, albeit not in the simplest way, as Fermat himself insisted on. And now those who wish can print their own versions.

However, the "fermist" family is not going to accept Wiles' proof at all. 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, "finally putting an end to a long-term epic."

By the way, just yesterday, one of the oldest "fermists" in our country, Vsevolod Yarosh, called the editorial office of "RG": "Do you know that I proved Fermat's theorem even before Wiles. Moreover, later I found a mistake in him, about which I wrote to our outstanding Mathematician Academician Arnold with a request to publish this in a scientific journal. Now I am waiting for an answer. I am also corresponding with the French Academy of Sciences on this matter. "

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

The editors of "RG" turned to the country's leading Institute of Mathematics. Steklov RAS with a request to evaluate this solution. Scientists were categorical: you can not 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 errors were made in the 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 at the Academy of Aviation and Aeronautics, Ilyin was supposed to present his proof. However, it turned out that few people even among specialists know about such an academy. And when, nevertheless, with great difficulty, it was possible to find the telephone number of the scientific secretary of this organization, then, as it turned out, he did not even suspect that such a historic event was to take place there. In a word, the correspondent of "RG" did not succeed in becoming a witness to the world sensation.

Once in the New Year's issue of the mailing list on how to make toasts, I casually mentioned that at the end of the 20th century there was one grandiose event that many did not notice - the so-called Fermat's Last Theorem was finally proved. On this occasion, among the letters I received, I found two responses from girls (one of them, as far as I remember, is Vika, a ninth-grader from Zelenograd), who were surprised by this fact.

And I was surprised by how keenly the girls are interested in the problems of modern mathematics. Therefore, I think that not only girls, but also boys of all ages - from high school students to pensioners, will also be interested in learning the history of the Great Theorem.

The proof of Fermat's theorem is a great event. And since it is not customary to joke with the word "great", then it seems to me that every self-respecting speaker (and all of us, when we say speakers) is simply obliged to know the history of the theorem.

If it so happened that you do not like mathematics as much as I love it, then look at some deepenings in detail with a cursory glance. Understanding that not all readers of our mailing list are interested in wandering in the wilds of mathematics, I tried not to give any formulas (except for the equation of Fermat's theorem and a couple of hypotheses) and to simplify the coverage of some specific issues as much as possible.

How Fermat brewed porridge

The French lawyer and part-time great mathematician of the 17th century, Pierre Fermat (1601-1665), put forward one curious statement from the field of number theory, which later became known as Fermat's Great (or Great) Theorem. This is one of the most famous and phenomenal mathematical theorems. Probably, the excitement around it would not have been so strong if in the book of Diophantus of Alexandria (3rd century AD) "Arithmetic", which Fermat often studied, making notes on its wide margins, and which his son Samuel kindly preserved for posterity , approximately the following entry of the great mathematician was not found:

"I have a very startling piece of evidence, but it's too big to fit in the margins."

It was this entry that caused the subsequent grandiose turmoil around the theorem.

So, the famous scientist said that he had proved his theorem. Let's ask ourselves the question: did he really prove it or did he lie corny? Or are there other versions explaining the appearance of that marginal entry that did not allow many mathematicians of the next generations to sleep peacefully?

The history of the Great Theorem is as fascinating as an adventure through time. Fermat stated in 1636 that an equation of the form x n + y n =z n has no solutions in integers with exponent n>2. This is actually Fermat's Last Theorem. In this seemingly simple mathematical formula, the Universe has masked incredible complexity. The Scottish-born American mathematician Eric Temple Bell, in his book The Final Problem (1961), even suggested that perhaps humanity would cease to exist before it could prove Fermat's Last Theorem.

It is somewhat strange that for some reason the theorem was late with its birth, since the situation was long overdue, because its special case for n = 2 - another famous mathematical formula - the Pythagorean theorem, arose twenty-two centuries earlier. Unlike Fermat's theorem, the Pythagorean theorem has an infinite number of integer solutions, for example, such Pythagorean triangles: (3,4,5), (5,12,13), (7,24,25), (8,15,17 ) … (27,36,45) … (112,384,400) … (4232, 7935, 8993) …

Grand Theorem Syndrome

Who just did not try to prove Fermat's theorem. Any fledgling student considered it his duty to apply to the Great Theorem, but no one was able to prove it. At first it didn't work for a hundred years. Then a hundred more. And further. A mass syndrome began to develop among mathematicians: "How is it? Fermat proved it, but what if I can't, or what?" - and some of them went crazy on this basis in the full sense of the word.

No matter how much the theorem was tested, it always turned out to be true. I knew one energetic programmer who was obsessed with the idea of ​​disproving the Great Theorem by trying to find at least one solution (counterexample) by iterating over integers using a fast computer (at that time more commonly called a computer). He believed in the success of his enterprise and liked to say: "A little more - and a sensation will break out!" I think that in different parts of our planet there were a considerable number of this kind of bold seekers. Of course, he did not find any solution. And no computers, even with fabulous speed, could ever test the theorem, because all the variables of this equation (including the exponents) can increase to infinity.

Theorem requires proof

Mathematicians know that if a theorem is not proven, anything (either true or false) can follow from it, as it did with some other hypotheses. For example, in one of his letters, Pierre Fermat suggested that numbers of the form 2 n +1 (the so-called Fermat numbers) are necessarily prime (that is, they do not have integer divisors and are divisible only by themselves and by one without a remainder), if n is a power of two (1, 2, 4, 8, 16, 32, 64, etc.). Fermat's hypothesis lived for more than a hundred years - until Leonhard Euler showed in 1732 that

2 32 +1 = 4 294 967 297 = 6 700 417 641

Then, almost 150 years later (1880), Fortune Landry factored the following Fermat number:

2 64 +1 = 18 446 744 073 709 551 617 = 274 177 67 280 421 310 721

How they could find the divisors of these large numbers without the help of computers - God only knows. In turn, Euler put forward the hypothesis that the equation x 4 + y 4 + z 4 =u 4 has no solutions in integers. However, about 250 years later, in 1988, Naum Elkis from Harvard managed to discover (already using a computer program) that

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

Therefore, Fermat's Last Theorem required proof, otherwise it was just a hypothesis, and it could well be that somewhere in the endless numerical fields the solution to the equation of the Great Theorem was lost.

The most virtuoso and prolific mathematician of the 18th century, Leonard Euler, whose archive of records mankind has been sorting out for almost a century, proved Fermat's theorem for powers 3 and 4 (or rather, he repeated the lost proofs of Pierre Fermat himself); his follower in number theory, Legendre (and independently Dirichlet) - for degree 5; Lame - for degree 7. But in general view the theorem remained unproved.

On March 1, 1847, at a meeting of the Paris Academy of Sciences, two outstanding mathematicians at once - Gabriel Lame and Augustin Cauchy - announced that they had come to the end of the proof of the Great Theorem and arranged a race, publishing their proofs in parts. However, the duel between them was interrupted because the same error was discovered in their proofs, which was pointed out by the German mathematician Ernst Kummer.

At the beginning of the 20th century (1908), a wealthy German entrepreneur, philanthropist and scientist Paul Wolfskel bequeathed one hundred thousand marks to anyone who would present a complete proof of Fermat's theorem. Already in the first year after the publication of Wolfskell's will by the Göttingen Academy of Sciences, it was inundated with thousands of proofs from lovers of mathematics, and this stream did not stop for decades, but, as you can imagine, they all contained errors. They say that the academy prepared forms with the following content:

Dear __________________________!
In your proof of Fermat's Theorem on ____ page ____ line from the top
The following error was found in the formula:__________________________:,

Which were sent to unlucky applicants for the award.

At that time, a semi-contemptuous nickname appeared in the circle of mathematicians - fermist. This was the name given to any self-confident upstart who lacked knowledge, but more than had ambition to hastily try his hand at proving the Great Theorem, and then, not noticing his own mistakes, proudly slapping his chest, loudly declare: "I proved the first Fermat's Theorem! Every farmer, even if he was ten thousandth in number, considered himself the first - this was ridiculous. The simple appearance of the Great Theorem reminded Fermists of easy prey so much that they were not at all embarrassed that even Euler and Gauss could not cope with it.

(Fermists, oddly enough, still exist today. Although one of them did not believe that he had proved the theorem like a classical fermist, but until recently he made attempts - he refused to believe me when I told him that Fermat's theorem had already been proved).

The most powerful mathematicians, perhaps in the quiet of their offices, also tried to cautiously approach this heavy barbell, but did not talk about it aloud, so as not to be branded as Fermists and, thus, not to harm their high authority.

By that time, the proof of the theorem for the exponent n appeared<100. Потом для n<619. Надо ли говорить о том, что все доказательства невероятно сложны. Но в общем виде теорема оставалась недоказанной.

Strange hypothesis

Until the middle of the twentieth century, no major advances in the history of the Great Theorem were observed. But soon an interesting event took place in mathematical life. In 1955, 28-year-old Japanese mathematician Yutaka Taniyama advanced a statement from a completely different area of ​​mathematics, called the Taniyama Hypothesis (aka the Taniyama-Shimura-Weil Hypothesis), which, unlike Fermat's belated Theorem, was ahead of its time.

Taniyama's conjecture states: "to every elliptic curve there corresponds a certain modular form." This statement for mathematicians of that time sounded about as absurd as the statement sounds for us: "a certain metal corresponds to each tree." It is easy to guess how a normal person can relate to such a statement - he simply will not take it seriously, which happened: mathematicians unanimously ignored the hypothesis.

A little explanation. Elliptic curves, known for a long time, have a two-dimensional form (located on a plane). Modular functions, discovered in the 19th century, have a four-dimensional form, so we cannot even imagine them with our three-dimensional brains, but we can describe them mathematically; in addition, modular forms are amazing in that they have the utmost possible symmetry - they can be translated (shifted) in any direction, mirrored, fragments can be swapped, rotated in infinitely many ways - and their appearance does not change. As you can see, elliptic curves and modular forms have little in common. Taniyama's hypothesis states that the descriptive equations of these two absolutely different mathematical objects corresponding to each other can be expanded into the same mathematical series.

Taniyama's hypothesis was too paradoxical: it combined completely different concepts - rather simple flat curves and unimaginable four-dimensional shapes. This never occurred to anyone. When, at an international mathematical symposium in Tokyo in September 1955, Taniyama demonstrated several correspondences between elliptic curves and modular forms, everyone saw this as nothing more than a funny coincidence. To Taniyama's modest question: is it possible to find the corresponding modular function for each elliptic curve, the venerable Frenchman Andre Weil, who at that time was one of the world's best specialists in number theory, gave a quite diplomatic answer, what, they say, if the inquisitive Taniyama does not leave enthusiasm, then maybe he will be lucky and his incredible hypothesis will be confirmed, but this must not happen soon. In general, like many other outstanding discoveries, at first Taniyama's hypothesis was ignored, because they had not grown up to it yet - almost no one understood it. Only one colleague of Taniyama, Goro Shimura, knowing his highly gifted friend well, intuitively felt that his hypothesis was correct.

Three years later (1958), Yutaka Taniyama committed suicide (however, samurai traditions are strong in Japan). From the point of view of common sense - an incomprehensible act, especially when you consider that very soon he was going to get married. The leader of young Japanese mathematicians began his suicide note as follows: “Yesterday I did not think about suicide. Recently, I often heard from others that I was mentally and physically tired. Actually, I still don’t understand why I’m doing this ...” and so on on three sheets. It’s a pity, of course, that this was the fate of an interesting person, but all geniuses are a little strange - that’s why they are geniuses (for some reason, the words of Arthur Schopenhauer came to mind: “in ordinary life, a genius is as much use as a telescope in a theater”) . The hypothesis has been abandoned. Nobody knew how to prove it.

For ten years, Taniyama's hypothesis was hardly mentioned. But in the early 70s, it became popular - it was regularly checked by everyone who could understand it - and it was always confirmed (as, in fact, Fermat's theorem), but, as before, no one could prove it.

The amazing connection between the two hypotheses

Another 15 years have passed. In 1984, there was one key event in the life of mathematics that combined the extravagant Japanese conjecture with Fermat's Last Theorem. The German Gerhard Frey put forward a curious statement, similar to a theorem: "If Taniyama's conjecture is proved, then, consequently, Fermat's Last Theorem will be proved." In other words, Fermat's theorem is a consequence of Taniyama's conjecture. (Frey, using ingenious mathematical transformations, reduced Fermat's equation to the form of an elliptic curve equation (the same one that appears in Taniyama's hypothesis), more or less substantiated his assumption, but could not prove it). And just a year and a half later (1986), a professor at the University of California, Kenneth Ribet, clearly proved Frey's theorem.

What happened now? Now it turned out that, since Fermat's theorem is already exactly a consequence of Taniyama's conjecture, all that is needed is to prove the latter in order to break the laurels of the conqueror of the legendary Fermat's theorem. But the hypothesis turned out to be difficult. In addition, over the centuries, mathematicians became allergic to Fermat's theorem, and many of them decided that it would also be almost impossible to cope with Taniyama's conjecture.

The death of Fermat's hypothesis. The birth of a theorem

Another 8 years have passed. One progressive English professor of mathematics from Princeton University (New Jersey, USA), Andrew Wiles, thought he had found a proof of Taniyama's conjecture. If the genius is not bald, then, as a rule, disheveled. Wiles is disheveled, therefore, looks like a genius. Entering into History, of course, is tempting and very desirable, but Wiles, like a real scientist, did not flatter himself, realizing that thousands of Fermists before him also saw ghostly evidence. Therefore, before presenting his proof to the world, he carefully checked it himself, but realizing that he could have a subjective bias, he also involved others in the checks, for example, under the guise of ordinary mathematical tasks, he sometimes threw various fragments of his proof to smart graduate students. Wiles later admitted that no one but his wife knew he was working on proving the Great Theorem.

And so, after long checks and painful reflections, Wiles finally plucked up courage, or, as he himself thought, impudence, and on June 23, 1993, at a mathematical conference on number theory in Cambridge, he announced his great achievement.

It was, of course, a sensation. No one expected such agility from a little-known mathematician. Then the press came along. Everyone was tormented by a burning interest. Slender formulas, like the strokes of a beautiful picture, appeared before the curious eyes of the audience. Real mathematicians, after all, they are like that - they look at all sorts of equations and see in them not numbers, constants and variables, but they hear music, like Mozart looking at a musical staff. Just like when we read a book, we look at the letters, but we don’t seem to notice them, but immediately perceive the meaning of the text.

The presentation of the proof seemed to be successful - no errors were found in it - no one heard a single false note (although most mathematicians simply stared at him like first graders at an integral and did not understand anything). Everyone decided that a large-scale event had happened: Taniyama's hypothesis was proved, and consequently Fermat's Last Theorem. But about two months later, a few days before the manuscript of Wiles's proof was to go into circulation, it was found to be inconsistent (Katz, a colleague of Wiles, noted that one piece of reasoning relied on "Euler's system", but what built by Wiles, was not such a system), although, in general, Wiles's techniques were considered interesting, elegant and innovative.

Wiles analyzed the situation and decided that he had lost. One can imagine how he felt with all his being what it means "from the great to the ridiculous one step." "I wanted to enter History, but instead I joined a team of clowns and comedians - arrogant farmists" - approximately such thoughts exhausted him during that painful period of his life. For him, a serious mathematician, it was a tragedy, and he threw his proof on the back burner.

But a little over a year later, in September 1994, while thinking about that bottleneck of the proof, together with his colleague Taylor from Oxford, the latter suddenly had the idea that the “Euler system” could be changed to the Iwasawa theory (section of number theory). Then they tried to use the Iwasawa theory, doing without the "Euler system", and they all came together. The corrected version of the proof was submitted for verification, and a year later it was announced that everything in it was absolutely clear, without a single mistake. In the summer of 1995, in one of the leading mathematical journals - "Annals of Mathematics" - a complete proof of Taniyama's conjecture (hence, Fermat's Great (Large) Theorem) was published, which occupied the entire issue - over one hundred sheets. The proof is so complex that only a few dozen people around the world could understand it in its entirety.

Thus, at the end of the 20th century, the whole world recognized that in the 360th year of its life, Fermat's Last Theorem, which in fact had been a hypothesis all this time, had become a proven theorem. Andrew Wiles proved Fermat's Great (Great) Theorem and entered History.

Think you've proven a theorem...

The happiness of the discoverer always goes to someone alone - it is he who, with the last blow of the hammer, cracks the hard nut of knowledge. But one cannot ignore the many previous blows that have formed a crack in the Great Theorem for centuries: Euler and Gauss (the kings of mathematics of their time), Evariste Galois (who managed to establish the theory of groups and fields in his short 21-year life, whose works were recognized as brilliant only after his death), Henri Poincaré (the founder of not only bizarre modular forms, but also conventionalism - a philosophical trend), David Gilbert (one of the strongest mathematicians of the twentieth century), Yutaku Taniyama, Goro Shimura, Mordell, Faltings, Ernst Kummer, Barry Mazur, Gerhard Frey, Ken Ribbet, Richard Taylor and others real scientists(I'm not afraid of these words).

The proof of Fermat's Last Theorem can be put on a par with such achievements of the twentieth century as the invention of the computer, the nuclear bomb, and space flight. Although not so widely known about it, because it does not invade the zone of our momentary interests, such as a TV set or an electric light bulb, it was a flash of a supernova, which, like all immutable truths, will always shine on humanity.

You can say: "Just think, you proved some kind of theorem, who needs it?". A fair question. David Gilbert's answer will fit exactly here. When, to the question: "what is the most important task for science now?", He answered: "to catch a fly on the far side of the moon", he was reasonably asked: "but who needs it?", he replied like this:" Nobody needs it. But think about how many important and difficult problems need to be solved in order to accomplish this. "Think about how many problems mankind has been able to solve in 360 years before proving Fermat's theorem. In search of its proof, almost half of modern mathematics was discovered. We must also take into account that mathematics is the avant-garde of science (and, by the way, the only one of the sciences that is built without a single mistake), and any scientific achievements and inventions begin here. ".

* * *

And now let's go back to the beginning of our story, remember Pierre Fermat's entry in the margins of Diophantus's textbook and once again ask ourselves the question: did Fermat really prove his theorem? Of course, we cannot know this for sure, and as in any case, different versions arise here:

Version 1: Fermat proved his theorem. (To the question: "Did Fermat have exactly the same proof of his theorem?", Andrew Wiles remarked: "Fermat could not have so proof. This is the proof of the 20th century. "We understand that in the 17th century mathematics, of course, was not the same as at the end of the 20th century - in that era, d, Artagnan, the queen of sciences, did not yet possess those discoveries (modular forms, Taniyama's theorems , Freya, etc.), which only made it possible to prove Fermat's Last Theorem. Of course, one can assume: what the hell is not joking - what if Fermat guessed in a different way? This version, although probable, is practically impossible according to the majority of mathematicians);
Version 2: It seemed to Pierre de Fermat that he had proved his theorem, but there were errors in his proof. (That is, Fermat himself was also the first Fermatist);
Version 3: Fermat did not prove his theorem, but simply lied in the margins.

If one of the last two versions is correct, which is most likely, then a simple conclusion can be drawn: great people, although they are great, they can also make mistakes or sometimes do not mind lying(basically, this conclusion will be useful for those who are inclined to completely trust their idols and other rulers of thoughts). Therefore, when reading the works of authoritative sons of mankind or listening to their pathetic speeches, you have every right to doubt their statements. (Please note that to doubt is not to reject).

Reprinting of article materials is possible only with obligatory links to the site (on the Internet - hyperlink) and to the author

That the Abel Prize in 2016 will go to Andrew Wiles for proving the Taniyama-Shimura conjecture for semistable elliptic curves and the proof following from this conjecture great theorem Farm. Currently, the premium is 6 million Norwegian kroner, that is, approximately 50 million rubles. According to Wiles, the award came as a "complete surprise" to him.

Fermat's theorem, proved more than 20 years ago, still attracts the attention of mathematicians. In part, this is due to its formulation, which is understandable even to a schoolboy: prove that for natural numbers n>2 there are no such triples of non-zero integers that a n + b n = c n . Pierre de Fermat wrote this expression in the margins of Diophantus' Arithmetic, with the remarkable caption "I have found a truly wonderful proof [of this assertion] for this, but the margins of the book are too narrow for it." Unlike most math tales, this one is real.

The presentation of the award is a great occasion to recall ten entertaining stories related to Fermat's theorem.

1.

Before Andrew Wiles proved Fermat's theorem, it was more properly called a conjecture, that is, Fermat's hypothesis. The fact is that a theorem is, by definition, an already proven statement. However, for some reason, just such a name stuck to this statement.

2.

If we put n = 2 in Fermat's theorem, then such an equation has infinitely many solutions. These solutions are called "Pythagorean triples". They got this name because they correspond to right-angled triangles, the sides of which are expressed by just such sets of numbers. You can generate Pythagorean triples using these three formulas (m 2 - n 2, 2mn, m 2 + n 2). These formulas should be substituted different meanings m and n, and the result will be the triples we need. The main thing here, however, is to make sure that the resulting numbers will be greater than zero - lengths cannot be expressed as negative numbers.

By the way, it is easy to see that if all the numbers in the Pythagorean triple are multiplied by some non-zero, we get a new Pythagorean triple. Therefore, it is reasonable to study triples in which the three numbers in the aggregate do not have a common divisor. The scheme that we have described makes it possible to obtain all such triples - this is by no means a simple result.

3.

On March 1, 1847, at a meeting of the Paris Academy of Sciences, two mathematicians at once - Gabriel Lame and Augustin Cauchy - announced that they were on the verge of proving a remarkable theorem. They ran a race to publish pieces of evidence. Most academics cheered for Lame, because Cauchy was a self-righteous, intolerant religious fanatic (and, of course, an absolutely brilliant part-time mathematician). However, the match was not destined to end - through his friend Joseph Liouville, the German mathematician Ernst Kummer informed the academicians that there was one and the same error in the proofs of Cauchy and Lame.

At school, it is proved that the decomposition of a number into prime factors only. Both mathematicians believed that if you look at the decomposition of integers already in the complex case, then this property - uniqueness - will be preserved. However, it is not.

It is noteworthy that if we consider only m + i n, then the decomposition is unique. Such numbers are called Gaussian. But Lame and Cauchy's work required factoring in cyclotomic fields. These are, for example, numbers in which m and n are rational, and i satisfies the property i^k = 1.

4.

Fermat's theorem for n = 3 has a clear geometric meaning. Let's imagine that we have many small cubes. Suppose we have collected two large cubes from them. In this case, of course, the sides will be integers. Is it possible to find two such large cubes that, having disassembled them into their component small cubes, we could assemble one large cube from them? Fermat's Theorem says that this can never be done. It's funny that if you ask the same question for three cubes, the answer is yes. For example, there is such a quadruple of numbers, discovered by the wonderful mathematician Srinivas Ramanujan:

3 3 + 4 3 + 5 3 = 6 3

5.

Leonhard Euler was noted in the history of Fermat's theorem. He did not really succeed in proving the statement (or even approaching the proof), but he formulated the hypothesis that the equation

x 4 + y 4 + z 4 = u 4

has no solution in integers. All attempts to find a direct solution to such an equation turned out to be fruitless. It wasn't until 1988 that Nahum Elkies of Harvard managed to find a counterexample. It looks like this:

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

Usually this formula is remembered in the context of a numerical experiment. As a rule, in mathematics it looks like this: there is some formula. The mathematician checks this formula in simple cases, convinces himself of the truth and formulates some hypothesis. Then he (although more often some of his graduate students or students) writes a program in order to check that the formula is correct for sufficiently large numbers that cannot be counted by hand (we are talking about one such experiment with prime numbers). This is not a proof, of course, but an excellent reason to declare a hypothesis. All these constructions are based on the reasonable assumption that if there is a counterexample to some reasonable formula, then we will find it quickly enough.

Euler's conjecture reminds us that life is much more diverse than our fantasies: the first counterexample can be arbitrarily large.

6.

In fact, of course, Andrew Wiles wasn't trying to prove Fermat's Theorem - he was solving a more difficult problem called the Taniyama-Shimura conjecture. There are two remarkable classes of objects in mathematics. The first one is called modular forms and is essentially a function on the Lobachevsky space. These functions do not change during the movements of this very plane. The second is called "elliptic curves" and is the curves given by the equation of the third degree in the complex plane. Both objects are very popular in number theory.

In the 1950s, two talented mathematicians Yutaka Taniyama and Goro Shimura met in the library of the University of Tokyo. At that time, there was no special mathematics at the university: it simply did not have time to recover after the war. As a result, scientists studied using old textbooks and analyzed problems at seminars that in Europe and the USA were considered solved and not particularly relevant. It was Taniyama and Shimura who discovered that there is a correspondence between modular forms and elliptic functions.

They tested their conjecture on some simple classes of curves. It turned out that it works. So they suggested that this connection is always there. This is how the Taniyama-Shimura hypothesis appeared, and three years later Taniyama committed suicide. In 1984, the German mathematician Gerhard Frey showed that if Fermat's Theorem is wrong, then the Taniyama-Shimura conjecture is wrong. It followed from this that the one who proved this conjecture would also prove the theorem. And that's exactly what Wiles did - though not in a very general way.

7.

Wiles spent eight years proving the conjecture. And during the check, the reviewers found an error in it, which “killed” most of the proof, nullifying all the years of work. One of the reviewers, by the name of Richard Taylor, undertook to repair the hole with Wiles. While they were working, a message appeared that Elkies, the same one who found a counterexample to Euler's conjecture, also found a counterexample to Fermat's theorem (later it turned out that this was an April Fool's joke). Wiles fell into a depression and did not want to continue - the hole in the evidence could not be closed in any way. Taylor talked Wiles into wrestling for another month.

A miracle happened and by the end of the summer mathematicians managed to make a breakthrough - this is how the works "Modular elliptic curves and Fermat's Last Theorem" by Andrew Wiles (pdf) and "Ring-theoretic properties of some Hecke algebras" by Richard Taylor and Andrew Wiles were born. This was the correct proof. It was published in 1995.

8.

In 1908, the mathematician Paul Wolfskel died in Darmstadt. After himself, he left a will in which he gave the mathematical community 99 years to find a proof of Fermat's Last Theorem. The author of the proof should have received 100 thousand marks (by the way, the author of the counterexample would not have received anything). According to a popular legend, it was love that prompted the Wolfskell mathematicians to make such a gift. Here is how Simon Singh describes the legend in his book Fermat's Last Theorem:

The story begins with Wolfskehl becoming infatuated with a beautiful woman whose identity has never been established. Much to Wolfskel's regret, the mysterious woman rejected him. He fell into such deep despair that he decided to commit suicide. Wolfskel was a passionate man, but not impulsive, and therefore began to work out his death in every detail. He set a date for his suicide and decided to shoot himself in the head with the first strike of the clock at exactly midnight. During the remaining days, Wolfskel decided to put his affairs in order, which were going great, and on the last day he made a will and wrote letters to close friends and relatives.

Wolfskehl worked so diligently that he finished all his business before midnight and, in order to somehow fill the remaining hours, he went to the library, where he began to look through mathematical journals. He soon came across Kummer's classic paper explaining why Cauchy and Lame had failed. Kummer's work was one of the most significant mathematical publications of its century, and was the best reading for a mathematician contemplating suicide. Wolfskel carefully, line by line, followed Kummer's calculations. Unexpectedly, it seemed to Wolfskel that he had discovered a gap: the author made a certain assumption and did not substantiate this step in his reasoning. Wolfskehl wondered if he had really found a serious gap, or if Kummer's assumption was justified. If a gap was found, then there was a chance that Fermat's Last Theorem could be proved much easier than many thought.

Wolfskehl sat down at the table, carefully analyzed the “flawed” part of Kummer’s reasoning and began to sketch out a mini-proof, which was supposed to either support Kummer’s work or demonstrate the fallacy of the assumption he made and, as a result, refute all his arguments. By dawn, Wolfskehl had finished his calculations. The bad (mathematically) news was that Kummer's proof had been healed, and Fermat's Last Theorem was still out of reach. But there was good news: the time for suicide had passed, and Wolfskehl was so proud that he had managed to find and fill a gap in the work of the great Ernest Kummer that his despair and sadness dispelled themselves. Mathematics gave him back the thirst for life.

However, there is an alternative version. According to her, Wolfskel took up mathematics (and, in fact, Fermat's theorem) because of progressive multiple sclerosis, which prevented him from doing what he loved - being a doctor. And he left the money to mathematicians so as not to leave his wife, whom he simply hated by the end of his life.

9.

Attempts to prove Fermat's theorem by elementary methods led to the emergence of a whole class strange people called "fermatists". They were engaged in the fact that they produced a huge amount of evidence and did not despair at all when they found an error in these proofs.

At the Faculty of Mechanics and Mathematics of Moscow State University there was a legendary character named Dobretsov. He collected certificates from various departments and, using them, penetrated the Mekhmat. This was done solely in order to find the victim. Somehow he came across a young graduate student (the future academician Novikov). He, in his naivety, began to carefully study the stack of papers that Dobretsov slipped him with the words, they say, here is the proof. After another "here's a mistake ..." Dobretsov took the stack and stuffed it into his briefcase. From the second briefcase (yes, he walked around the mekhmat with two briefcases), he took out the second pile, sighed and said: "Well, then let's see option 7 B."

By the way, most of these proofs begin with the phrase "Let's move one of the terms to the right side of the equality and factorize it."

10.

The story about the theorem would be incomplete without the wonderful film "The Mathematician and the Devil".

Amendment

Section 7 of this paper originally stated that Naum Elkies had found a counterexample to Fermat's theorem, which later turned out to be wrong. This is not true: the message about the counterexample was an April Fool's joke. We apologize for the inaccuracy.

Andrey Konyaev

Ivliev Yu.A.

The article is devoted to the description of a fundamental mathematical error made in the process of proving Fermat's Last Theorem at the end of the 20th century. The detected error not only distorts the true meaning of the theorem, but also hinders the development of a new axiomatic approach to the study of powers of numbers and the natural series of numbers.

In 1995, an article was published that was similar in size to a book and reported on the proof of the famous Fermat's Great (Last) Theorem (WTF) (for the history of the theorem and attempts to prove it, see, for example,). After this event, many scientific articles and popular science books appeared that promote this proof, but none of these works revealed a fundamental mathematical error in it, which crept in not even through the fault of the author, but due to some strange optimism that gripped the minds mathematicians who dealt with this problem and related questions. The psychological aspects of this phenomenon have been investigated in. It also gives a detailed analysis of the oversight that occurred, which is not of a particular nature, but is the result of an incorrect understanding of the properties of the powers of integers. As shown in , Fermat's problem is rooted in a new axiomatic approach to the study of these properties, which has not yet been applied in modern science. But an erroneous proof stood in his way, giving number theorists false guidelines and leading researchers of Fermat's problem away from its direct and adequate solution. This work is devoted to removing this obstacle.

1. Anatomy of a mistake made during the proof of the WTF

In the process of very long and tedious reasoning, Fermat's original statement was reformulated in terms of a correspondence between a Diophantine equation of the p-th degree and elliptic curves of the 3rd order (see Theorems 0.4 and 0.5 in ). Such a comparison forced the authors of the de facto collective proof to announce that their method and reasoning lead to the final solution of Fermat's problem (recall that the WTF did not have recognized proofs for the case of arbitrary integer powers of integers until the 90s of the last century). The purpose of this consideration is to establish the mathematical incorrectness of the above comparison and, as a result of the analysis, to find a fundamental error in the proof presented in .

a) Where and what is wrong?

So, let's go through the text, where on p.448 it is said that after the "witty idea" of G. Frey (G. Frey), the possibility of proving the WTF has opened up. In 1984, G. Frey suggested and

K.Ribet later proved that the putative elliptic curve representing the hypothetical integer solution of Fermat's equation,

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

cannot be modular. However, A.Wiles and R.Taylor proved that any semistable elliptic curve defined over the field of rational numbers is modular. This led to the conclusion about the impossibility of integer solutions of Fermat's equation and, consequently, the validity of Fermat's statement, which in the notation of A. Wiles was written as Theorem 0.5: let there be an equality

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

where u, v, w- rational numbers, integer exponent p ≥ 3; then (2) is satisfied only if uvw = 0 .

Now, apparently, we should go back and critically consider why the curve (1) was a priori perceived as elliptic and what is its real relationship with Fermat's equation. Anticipating this question, A. Wiles refers to the work of Y. Hellegouarch, in which he found a way to associate Fermat's equation (presumably solved in integers) with a hypothetical 3rd order curve. Unlike G. Frey, I. Allegouches did not connect his curve with modular forms, but his method of obtaining equation (1) was used to further advance the proof of A. Wiles.

Let's take a closer look at work. The author conducts his reasoning in terms of projective geometry. Simplifying some of its notation and bringing them into line with , we find that the Abelian curve

Y 2 = X(X - β p)(X + γ p) (3)

the Diophantine equation is compared

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

where x, y, z are unknown integers, p is an integer exponent from (2), and the solutions of the Diophantine equation (4) α p , β p , γ p are used to write the Abelian curve (3).

Now, to make sure that this is an elliptic curve of the 3rd order, it is necessary to consider the variables X and Y in (3) on the Euclidean plane. To do this, we use the well-known rule of arithmetic for elliptic curves: if there are two rational points on a cubic algebraic curve and the line passing through these points intersects this curve at one more point, then the latter is also a rational point. Hypothetical equation (4) formally represents the law of addition of points on a straight line. If we make a change of variables x p = A, y p=B, z p = C and direct the straight line thus obtained along the X axis in (3), then it will intersect the 3rd degree curve at three points: (X = 0, Y = 0), (X = β p , Y = 0), (X = - γ p , Y = 0), which is reflected in the notation of the Abelian curve (3) and in a similar notation (1). However, is curve (3) or (1) really elliptical? Obviously not, because the segments of the Euclidean line, when adding points on it, are taken on a non-linear scale.

Returning to the linear coordinate systems of the Euclidean space, instead of (1) and (3) we obtain formulas that are very different from the formulas for elliptic curves. For example, (1) could be of the following form:

η 2p = ξ p (ξ p + u p)(ξ p - v p) (5)

where ξ p = x, η p = y, and the appeal to (1) in this case for the derivation of the WTF seems to be illegal. Despite the fact that (1) satisfies some criteria of the class of elliptic curves, it does not satisfy the most important criterion of being a 3rd degree equation in a linear coordinate system.

b) Error classification

So, once again we return to the beginning of the consideration and follow how the conclusion about the truth of the WTF is made. First, it is assumed that there is a solution of Fermat's equation in positive integers. Secondly, this solution is arbitrarily inserted into an algebraic form of a known form (a plane curve of the 3rd degree) under the assumption that the elliptic curves obtained in this way exist (the second unconfirmed assumption). Thirdly, since it is proved by other methods that the constructed concrete curve is non-modular, it means that it does not exist. The conclusion follows from this: there is no integer solution of the Fermat equation and, therefore, the WTF is true.

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

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

How did it happen that such an elementary mistake was missed in a serious mathematical work? Probably, this happened due to the fact that “illusory” geometric figures of this type were not previously studied in mathematics. Indeed, who could be interested, for example, in a fictitious circle obtained from Fermat's equation by changing the variables x n/2 = A, y n/2 = B, z n/2 = C? After all, its equation C 2 = A 2 + B 2 has no integer solutions for integer x, y, z and n ≥ 3 . In non-linear coordinate axes X and Y, such a circle would be described by the equation, according to appearance very similar to the standard form:

Y 2 \u003d - (X - A) (X + B),

where A and B are no longer variables, but concrete numbers determined by the above substitution. But if the numbers A and B are given their original form, which consists in their power character, then the heterogeneity of the notation in the factors on the right side of the equation immediately catches the eye. This sign helps to distinguish illusion from reality and to move from non-linear to linear coordinates. On the other hand, if we consider numbers as operators when comparing them with variables, as for example in (1), then both must be homogeneous quantities, i.e. must have the same degree.

Such an understanding of the powers of numbers as operators also makes it possible to see that the comparison of Fermat's equation with an illusory elliptic curve is not unambiguous. Take, for example, one of the factors on the right side of (5) and expand it into p linear factors by introducing a complex number r such that r p = 1 (see for example ):

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

Then form (5) can be represented as a decomposition into prime factors of complex numbers according to the type of algebraic identity (6), however, the uniqueness of such a decomposition in general case is questionable, which was once shown by Kummer.

2. Conclusions

It follows from the previous analysis that the so-called arithmetic of elliptic curves is not capable of shedding light on where to look for the proof of the WTF. After the work, Fermat's statement, by the way, taken as the epigraph to this article, began to be perceived as a historical joke or practical joke. However, in reality it turns out that it was not Fermat who was joking, but the experts who gathered at the mathematical symposium in Oberwolfach in Germany in 1984, at which G. Frey voiced his witty idea. The consequences of such a careless statement brought mathematics as a whole to the verge of losing its public trust, which is described in detail in and which necessarily raises the question of the responsibility of scientific institutions to society before science. The mapping of the Fermat equation to the Frey curve (1) is the "lock" of Wiles's entire proof with respect to Fermat's theorem, and if there is no correspondence between the Fermat curve and modular elliptic curves, then there is no proof either.

Lately there have been various Internet reports that some prominent mathematicians have finally figured out Wiles' proof of Fermat's theorem, giving him an excuse in the form of a "minimal" recalculation of integer points in Euclidean space. However, no innovations can cancel the classical results already obtained by mankind in mathematics, in particular, the fact that although any ordinal number coincides with its quantitative counterpart, it cannot be a replacement for it in operations of comparing numbers with each other, and hence with inevitably follows the conclusion that the Frey curve (1) is not elliptic initially, i.e. is not by definition.

BIBLIOGRAPHY:

1. Ivliev Yu.A. Reconstruction of the native proof of Fermat's Last Theorem - United Scientific Journal (section "Mathematics"). April 2006 No. 7 (167) p.3-9, see also Pratsi of the Luhansk branch of the International Academy of Informatization. Ministry of Education and Science of Ukraine. Shidnoukrainian National University named after. V. Dahl. 2006 No. 2 (13) pp.19-25.
2. Ivliev Yu.A. The greatest scientific scam of the 20th century: the "proof" of Fermat's Last Theorem - Natural and technical sciences (section "History and methodology of mathematics"). August 2007 No. 4 (30) pp. 34-48.
3. Edwards G. (Edwards H.M.) Fermat's last theorem. Genetic introduction to algebraic number theory. Per. from English. ed. B.F. Skubenko. M.: Mir 1980, 484 p.
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 link