Newton's experiment on dispersion. Newton's color experiments. Displays a continuous spectrum of white light

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, Great Theorem Fermat (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.

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 a square built on the hypotenuse is equal to the sum of squares built on the legs.

In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triplets satisfying the equality x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas to find them. They probably tried to look for 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 is difficult.

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


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

But the 17th century mathematician Frenchman Pierre de Fermat enthusiastically 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-1980s it became clear that scientific world is on the way to the final solution of Fermat's Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat's last theorem was practically over.

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

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

Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that 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 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 unexpectedly gave mathematicians new method 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...

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Grigory Perelman. refusenik

Vasily Maksimov

In August 2006, the names of the best mathematicians on the planet were announced who received the prestigious Fields Medal - a kind of analogue of the Nobel Prize, which mathematicians, at the whim of Alfred Nobel, were deprived of. The Fields Medal - in addition to a badge of honor, the winners are awarded a check for fifteen thousand Canadian dollars - is awarded by the International Congress of Mathematicians every four years. It was established by Canadian scientist John Charles Fields and was first awarded in 1936. Since 1950, the Fields Medal has been awarded regularly personally by the King of Spain for his contribution to the development of mathematical science. Prize winners can be from one to four scientists under the age of forty. Forty-four mathematicians, including eight Russians, have already received the prize.

Grigory Perelman. Henri Poincare.

In 2006, the laureates were the Frenchman Wendelin Werner, the Australian Terence Tao and two Russians - Andrey Okunkov working in the USA and Grigory Perelman, a scientist from St. Petersburg. However, at the last moment it became known that Perelman refused this prestigious award - as the organizers announced, “for reasons of principle.”

Such an extravagant act by the Russian mathematician did not come as a surprise to people who knew him. This is not the first time he has refused mathematical awards, explaining his decision by saying that he does not like ceremonial events and unnecessary hype around his name. Ten years ago, in 1996, Perelman refused the European Mathematical Congress prize, citing the fact that he had not completed the work on the scientific problem nominated for the award, and this was not the last case. The Russian mathematician seemed to make it his life’s goal to surprise people, going against public opinion and the scientific community.

Grigory Yakovlevich Perelman was born on June 13, 1966 in Leningrad. From a young age I was interested in exact sciences, graduated with distinction from the famous 239th high school With in-depth study mathematics, won numerous mathematical Olympiads: for example, in 1982, as part of a team of Soviet schoolchildren, he participated in the International Mathematical Olympiad, held in Budapest. Without exams, Perelman was enrolled in the Faculty of Mechanics and Mathematics at Leningrad University, where he studied with excellent marks, continuing to win mathematical competitions at all levels. After graduating from the university with honors, he entered graduate school at the St. Petersburg branch of the Steklov Mathematical Institute. His scientific supervisor was the famous mathematician Academician Aleksandrov. Protecting candidate's thesis, Grigory Perelman remained at the institute, in the laboratory of geometry and topology. His work on the theory of Alexandrov spaces is known; he was able to find evidence for a number of important conjectures. Despite numerous offers from leading Western universities, Perelman prefers to work in Russia.

His most notable success was the solution in 2002 of the famous Poincaré conjecture, published in 1904 and since then remained unproven. Perelman worked on it for eight years. The Poincaré conjecture was considered one of the greatest mathematical mysteries, and its solution was considered the most important achievement in mathematical science: it would immediately advance research into the problems of the physical and mathematical foundations of the universe. The most prominent minds on the planet predicted its solution only in a few decades, and the Clay Institute of Mathematics in Cambridge, Massachusetts, included the Poincaré problem among the seven most interesting unsolved mathematical problems of the millennium, for the solution of each of which a million dollar prize was promised (Millennium Prize Problems). .

The conjecture (sometimes called the problem) of the French mathematician Henri Poincaré (1854–1912) is formulated as follows: any closed simply connected three-dimensional space is homeomorphic to a three-dimensional sphere. To clarify, use a clear example: if you wrap an apple with a rubber band, then, in principle, by tightening the tape, you can compress the apple into a point. If you wrap a donut with the same tape, you cannot compress it to a point without tearing either the donut or the rubber. In this context, an apple is called a “simply connected” figure, but a donut is not simply connected. Almost a hundred years ago, Poincaré established that a two-dimensional sphere is simply connected, and suggested that a three-dimensional sphere is also simply connected. The best mathematicians in the world could not prove this hypothesis.

To qualify for the Clay Institute Prize, Perelman only had to publish his solution in one of the scientific journals, and if within two years no one could find an error in his calculations, then the solution would be considered correct. However, Perelman deviated from the rules from the very beginning, publishing his decision on the preprint website of the Los Alamos Scientific Laboratory. Perhaps he was afraid that an error had crept into his calculations - a similar story had already happened in mathematics. In 1994, the English mathematician Andrew Wiles proposed a solution to Fermat’s famous theorem, and a few months later it turned out that an error had crept into his calculations (although it was later corrected, and the sensation still took place). There is still no official publication of the proof of the Poincaré conjecture, but there is an authoritative opinion of the best mathematicians on the planet confirming the correctness of Perelman’s calculations.

The Fields Medal was awarded to Grigory Perelman precisely for solving the Poincaré problem. But the Russian scientist refused the prize, which he undoubtedly deserves. “Gregory told me that he feels isolated from the international mathematical community, outside this community, and therefore does not want to receive the award,” Englishman John Ball, president of the World Union of Mathematicians (WUM), said at a press conference in Madrid.

There are rumors that Grigory Perelman is going to leave science altogether: six months ago he resigned from his native Steklov Mathematical Institute, and they say that he will no longer study mathematics. Perhaps the Russian scientist believes that by proving the famous hypothesis, he has done everything he could for science. But who will undertake to discuss the train of thought of such a bright scientist and extraordinary person?.. Perelman refuses any comments, and he told The Daily Telegraph newspaper: “None of what I can say is of the slightest public interest.” However, leading scientific publications were unanimous in their assessments when they reported that “Grigory Perelman, having resolved the Poincaré theorem, stood on a par with the greatest geniuses of the past and present.”

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