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 its essence and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n > 2. Everything seems to be simple and clear, but the best mathematicians and simple amateurs fought over searching for a solution for more than three and a half centuries.
Why is she so famous? Now let's find out...
Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with 5th grade high school, but the proof is not even any professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved 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 all sides are equal." 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 triples satisfying the equation x²+y²=z². They proved that Pythagorean triplets infinitely many, and got general formulas to find them. They probably tried to look for triples and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.
That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²
Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
Well, and so on. What if we take a similar equation x³+y³=z³ ? Maybe there are such numbers too?
And so on (Fig. 1).
Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.
It is more difficult to prove the absence: 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 a solution). And that's it, the opponent is defeated. How to prove absence?
To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.
In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2):
And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:
But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically studied the general equation x n+yn=zn . And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! 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 accommodate it."
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.
After Fermat, great minds such as Leonhard Euler worked on finding the proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a proof for n = 7) and many others. By the mid-1980s, it became clear that academia is on the way to the final solution of Fermat's Last Theorem, but it was not until 1993 that mathematicians saw and believed that the three-century saga of finding a proof of Fermat's Last Theorem was almost over.
It is easy to show that it suffices to prove Fermat's theorem only for prime 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, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.
Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that by the methods of mathematics of the 19th century, the theorem in general view cannot be proven. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.
In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.
He soon died of natural causes. The heirs were pretty 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 Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:
Dear (s). . . . . . . .
Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, drawing 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 unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups 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 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.
Nevertheless, 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 a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole 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 cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proved 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 proved. But for thirty years it was not possible to prove the Taniyama–Shimura conjecture, and there were less and less hopes for success.
In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.
Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama–Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat's Last Theorem causes too much great 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 report 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. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.
It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following 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 gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?
This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ...
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 its essence and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n > 2. Everything seems to be simple and clear, but the best mathematicians and simple amateurs fought over searching for a solution for more than three and a half centuries.
Why is she so famous? Now let's find out...
Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with the 5th grade of secondary school, but the proof is far from even every professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved 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 triples satisfying the equation 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 triples and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.
That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²
Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
Well, and so on. What if we take a similar equation x³+y³=z³ ? Maybe there are such numbers too?
And so on (Fig. 1).
Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.
It is more difficult to prove the absence: 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 a solution). And that's it, the opponent is defeated. How to prove absence?
To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.
In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2):
And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:
But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically studied the general equation x n+yn=zn . And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! 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 accommodate it."
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.
After Fermat, great minds such as Leonhard Euler worked on finding the proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a 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 did mathematicians see and believe that the three-century saga of finding a proof of Fermat's last theorem was almost over.
It is easy to show that it suffices to prove Fermat's theorem only for prime 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, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.
Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that the methods of mathematics of the 19th century cannot prove the theorem in general form. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.
In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.
He soon died of natural causes. The heirs were pretty 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 Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:
Dear (s). . . . . . . .
Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, drawing 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 unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups 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 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.
Nevertheless, 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 a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole 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 cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proved 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 proved. But for thirty years it was not possible to prove the Taniyama–Shimura conjecture, and there were less and less hopes for success.
In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.
Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama–Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat’s Last Theorem is of too much interest ... Too many viewers deliberately 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 report 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. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.
It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following 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 gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?
This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ... Fermat's interest in mathematics appeared somehow unexpectedly and at a fairly mature age. In 1629, a Latin translation of the work of Pappus, containing a brief summary of the results of Apollonius on the properties conic sections. Fermat, a polyglot, an expert in law and ancient philology, suddenly sets out to completely restore the course of reasoning of the famous scientist. With the same success, a modern lawyer can try to independently reproduce all the proofs from a monograph from problems, say, of algebraic topology. However, the unthinkable enterprise is crowned with success. Moreover, delving into the geometric constructions of the ancients, he makes an amazing discovery: in order to find the maxima and minima of the areas of figures, ingenious drawings are not needed. It is always possible to compose and solve some simple algebraic equation, the roots of which determine the extremum. He came up with an algorithm that would become the basis of differential calculus.
He quickly moved on. He found sufficient conditions for the existence of maxima, learned to determine the inflection points, drew tangents to all known curves of the second and third order. A few more years, and he finds a new purely algebraic method for finding quadratures for parabolas and hyperbolas of arbitrary order (that is, integrals of functions of the form y p = Cx q and y p x q \u003d C), calculates areas, volumes, moments of inertia of bodies of revolution. It was a real breakthrough. Feeling this, Fermat begins to seek communication with the mathematical authorities of the time. He is confident and longs for recognition.
In 1636 he wrote the first letter to His Reverend Marin Mersenne: “Holy Father! I am extremely grateful to you for the honor you have done me by giving me the hope that we will be able to talk in writing; ...I will be very glad to hear from you about all the new treatises and books on Mathematics that have appeared in the last five or six years. ...I also found a lot analytical methods for various problems, both numerical and geometric, for which Vieta's analysis is insufficient. All this I will share with you whenever you want, and, moreover, without any arrogance, from which I am freer and more distant than any other person in the world.
Who is Father Mersenne? This is a Franciscan monk, a scientist of modest talents and a wonderful organizer, who for 30 years headed the Parisian mathematical circle, which became the true center of French science. Subsequently, the Mersenne circle, by decree of Louis XIV, will be transformed into the Paris Academy of Sciences. Mersenne tirelessly carried on a huge correspondence, and his cell in the monastery of the Order of the Minims on the Royal Square was a kind of "post office for all the scientists of Europe, from Galileo to Hobbes." Correspondence then replaced scientific journals, which appeared much later. Meetings at Mersenne took place weekly. The core of the circle was made up of the most brilliant natural scientists of that time: Robertville, Pascal Father, Desargues, Midorge, Hardy and, of course, the famous and universally recognized Descartes. Rene du Perron Descartes (Cartesius), a mantle of nobility, two family estates, the founder of Cartesianism, the “father” of analytic geometry, one of the founders of new mathematics, as well as Mersenne’s friend and comrade at the Jesuit College. This wonderful man will be Fermat's nightmare.
Mersenne found Fermat's results interesting enough to bring the provincial into his elite club. The farm immediately strikes up a correspondence with many members of the circle and literally falls asleep with letters from Mersenne himself. In addition, he sends completed manuscripts to the court of pundits: “Introduction to flat and solid places”, and a year later - “The method of finding maxima and minima” and “Answers to B. Cavalieri's questions”. What Fermat expounded was absolutely new, but the sensation did not take place. Contemporaries did not flinch. They didn’t understand much, but they found unambiguous indications that Fermat borrowed the idea of the maximization algorithm from Johannes Kepler’s treatise with the funny title “The New Stereometry of Wine Barrels”. Indeed, in Kepler's reasoning there are phrases like “The volume of the figure is greatest if on both sides of the place the greatest value the decrease is at first insensitive.” But the idea of a small increment of a function near an extremum was not at all in the air. The best analytical minds of that time were not ready for manipulations with small quantities. The fact is that at that time algebra was considered a kind of arithmetic, that is, mathematics of the second grade, a primitive improvised tool developed for the needs of base practice (“only merchants count well”). Tradition prescribed to adhere to purely geometric methods of proofs, dating back to ancient mathematics. Fermat was the first to understand that infinitesimal quantities can be added and reduced, but it is rather difficult to represent them as segments.
It took almost a century for Jean d'Alembert to admit in his famous Encyclopedia: Fermat was the inventor of the new calculus. It is with him that we meet the first application of differentials for finding tangents.” AT late XVIII century, Joseph Louis Comte de Lagrange will express himself even more definitely: “But the geometers - Fermat's contemporaries - did not understand this new kind of calculus. They saw only special cases. And this invention, which appeared shortly before Descartes' Geometry, remained fruitless for forty years. Lagrange is referring to 1674, when Isaac Barrow's "Lectures" were published, covering Fermat's method in detail.
Among other things, it quickly became clear that Fermat was more inclined to formulate new problems than to humbly solve the problems proposed by the meters. In the era of duels, the exchange of tasks between pundits was generally accepted as a form of clarifying issues related to chain of command. However, the Farm clearly does not know the measure. Each of his letters is a challenge containing dozens of complex unsolved problems, and on the most unexpected topics. Here is an example of his style (addressed to Frenicle de Bessy): “Item, what is the smallest square that, when reduced by 109 and added to one, will give a square? If you do not send me the general solution, then send me the quotient for these two numbers, which I chose small so as not to make you very difficult. After I get your answer, I will suggest some other things to you. It is clear, without special reservations, that in my proposal it is required to find whole numbers, because in the case of fractional numbers, the most insignificant arithmetician could reach the goal.” Fermat often repeated himself, formulating the same questions several times, and openly bluffed, claiming that he had an unusually elegant solution to the proposed problem. There were no direct errors. Some of them were noticed by contemporaries, and some of the insidious statements misled readers for centuries.
Mersenne's circle reacted adequately. Only Robertville, the only member of the circle who had problems with the origin, maintains a friendly tone of letters. The good shepherd Father Mersenne tried to reason with the "Toulouse impudent". But Farm does not intend to make excuses: “Reverend Father! You write to me that the posing of my impossible problems angered and cooled Messrs. Saint-Martin and Frenicle, and that this was the reason for the termination of their letters. However, I want to object to them that what seems impossible at first is actually not, and that there are many problems that, as Archimedes said...” etc.
However, Farm is disingenuous. It was Frenicle who sent the problem of finding right triangle with integer sides, whose area is equal to the square of the integer. He sent it, although he knew that the problem obviously had no solution.
The most hostile position towards Fermat was taken by Descartes. In his letter to Mersenne dated 1938 we read: “because I found out that this is the same person who had previously tried to refute my “Dioptric”, and since you informed me that he sent it after he had read my “Geometry ” and in surprise that I did not find the same thing, i.e. (as I have reason to interpret it) sent it with the aim of entering into rivalry and showing that he knows more about it than I do, and since more of your letters, I learned that he had a reputation as a very knowledgeable geometer, then I consider myself obliged to answer him. Descartes will later solemnly designate his answer as “the small trial of Mathematics against Mr. Fermat”.
It is easy to understand what infuriated the eminent scientist. First, in Fermat's reasoning, coordinate axes and the representation of numbers by segments constantly appear - a device that Descartes comprehensively develops in his just published "Geometry". Fermat comes to the idea of replacing the drawing with calculations on his own, in some ways even more consistent than Descartes. Secondly, Fermat brilliantly demonstrates the effectiveness of his method of finding minima on the example of the problem of the shortest path of a light beam, refining and supplementing Descartes with his "Dioptric".
The merits of Descartes as a thinker and innovator are enormous, but let's open the modern "Mathematical Encyclopedia" and look at the list of terms associated with his name: "Cartesian coordinates" (Leibniz, 1692), "Cartesian sheet", "Descartes ovals". None of his arguments went down in history as Descartes' Theorem. Descartes is primarily an ideologist: he is the founder of a philosophical school, he forms concepts, improves the system of letter designations, but there are few new specific techniques in his creative legacy. In contrast, Pierre Fermat writes little, but on any occasion he can come up with a lot of witty mathematical tricks (see ibid. "Fermat's Theorem", "Fermat's Principle", "Fermat's method of infinite descent"). They probably quite rightly envied each other. The collision was inevitable. With the Jesuit mediation of Mersenne, a war broke out that lasted two years. However, Mersenne turned out to be right before history here too: the fierce battle between the two titans, their tense, to put it mildly, polemic contributed to the understanding of the key concepts of mathematical analysis.
Fermat is the first to lose interest in the discussion. Apparently, he spoke directly with Descartes and never again offended his opponent. In one of his recent works“Synthesis for refraction”, the manuscript of which he sent to de la Chaumbra, Fermat commemorates “the most learned Descartes” through the word and in every possible way emphasizes his priority in matters of optics. Meanwhile, it was this manuscript that contained the description of the famous "Fermat's principle", which provides an exhaustive explanation of the laws of reflection and refraction of light. Curtseys to Descartes in a work of this level were completely unnecessary.
What happened? Why did Fermat, putting aside pride, went to reconciliation? Reading Fermat's letters of those years (1638 - 1640), one can assume the simplest thing: during this period, his scientific interests changed dramatically. He abandons the fashionable cycloid, ceases to be interested in tangents and areas, and for a long 20 years forgets about his method of finding the maximum. Having great merits in the mathematics of the continuous, Fermat completely immerses himself in the mathematics of the discrete, leaving the hateful geometric drawings to his opponents. Numbers are his new passion. As a matter of fact, the entire "Theory of Numbers", as an independent mathematical discipline, owes its birth entirely to the life and work of Fermat.
<…>After Fermat's death, his son Samuel published in 1670 a copy of Arithmetic belonging to his father under the title "Six books of arithmetic by the Alexandrian Diophantus with comments by L. G. Basche and remarks by P. de Fermat, Senator of Toulouse." The book also included some of Descartes' letters and the full text of Jacques de Bigly's A New Discovery in the Art of Analysis, based on Fermat's letters. The publication was an incredible success. An unprecedented bright world opened up before the astonished specialists. The unexpectedness, and most importantly, the accessibility, democratic nature of Fermat's number-theoretic results gave rise to a lot of imitations. At that time, few people understood how the area of a parabola was calculated, but every student could understand the formulation of Fermat's Last Theorem. A real hunt began for the unknown and lost letters of the scientist. Before late XVII in. Every word of his that was found was published and republished. But the turbulent history of the development of Fermat's ideas was just beginning.
Judging by the popularity of the query "Fermat's theorem - short proof, this mathematical problem is really of interest to many. This theorem was first stated by Pierre de Fermat in 1637 on the edge of a copy of the Arithmetic, where he claimed that he had a solution that was too large to fit on the edge.
The first successful proof was published in 1995 - it was complete proof Fermat's Theorem by Andrew Wiles. It has been described as "staggering progress" and led Wiles to receive the Abel Prize in 2016. Although described relatively briefly, the proof of Fermat's theorem also proved much of the modularity theorem and opened up new approaches to numerous other problems and effective methods the rise of modularity. These accomplishments have advanced mathematics 100 years into the future. The proof of Fermat's little theorem today is not something out of the ordinary.
The unresolved problem stimulated the development of algebraic number theory in the 19th century and the search for a proof of the modularity theorem in the 20th century. This is one of the most notable theorems in the history of mathematics, and until the full proof of Fermat's Last Theorem by division, it was in the Guinness Book of Records as "the most difficult mathematical problem", one of the features of which is that it has the largest number bad evidence.
History reference
The Pythagorean equation x 2 + y 2 = z 2 has an infinite number of positive integer solutions for x, y and z. These solutions are known as Pythagorean trinities. Around 1637, Fermat wrote on the edge of the book that the more general equation a n + b n = c n has no solutions in natural numbers, if n is an integer greater than 2. Although Fermat himself claimed to have a solution to his problem, he left no details about its proof. The elementary proof of Fermat's theorem, claimed by its creator, was rather his boastful invention. The book of the great French mathematician was discovered 30 years after his death. This equation, called Fermat's Last Theorem, remained unsolved in mathematics for three and a half centuries.
The theorem eventually became one of the most notable unsolved problems in mathematics. Attempts to prove this caused a significant development in number theory, and over time Fermat's last theorem became known as an unsolved problem in mathematics.
A Brief History of the Evidence
If n = 4, as proved by Fermat himself, it suffices to prove the theorem for indices n that are prime numbers. Over the next two centuries (1637-1839) the conjecture was only proven for the primes 3, 5 and 7, although Sophie Germain updated and proved an approach that applied to the whole class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, whereby irregular primes were analyzed individually. Based on Kummer's work and using sophisticated computer research, other mathematicians were able to extend the solution of the theorem, with the goal of covering all the main exponents up to four million, but the proof for all exponents was still not available (meaning that mathematicians usually considered the solution of the theorem impossible, extremely difficult, or unattainable with modern knowledge).
The work of Shimura and Taniyama
In 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected that there was a connection between elliptic curves and modular forms, two very different branches of mathematics. Known at the time as the Taniyama-Shimura-Weil conjecture and (ultimately) as the modularity theorem, it existed on its own, with no apparent connection to Fermat's last theorem. It itself was widely regarded as an important mathematical theorem, but it was considered (like Fermat's theorem) impossible to prove. At the same time, the proof of Fermat's Last Theorem (by dividing and applying complex mathematical formulas) was not completed until half a century later.
In 1984, Gerhard Frey noticed an obvious connection between these two previously unrelated and unresolved problems. A complete confirmation that the two theorems were closely related was published in 1986 by Ken Ribet, who based on a partial proof by Jean-Pierre Serra, who proved all but one part, known as the "epsilon hypothesis". Simply put, these works by Frey, Serra, and Ribe showed that if the modularity theorem could be proved, at least for a semistable class of elliptic curves, then the proof of Fermat's last theorem would sooner or later be discovered as well. Any solution that can contradict Fermat's last theorem can also be used to contradict the modularity theorem. Therefore, if the modularity theorem turned out to be true, then by definition there cannot be a solution that contradicts Fermat's last theorem, which means that it should have been proved soon.
Although both theorems were hard problems in mathematics, considered unsolvable, the work of the two Japanese was the first suggestion of how Fermat's last theorem could be extended and proved for all numbers, not just some. Important for the researchers who chose the research topic was the fact that, unlike Fermat's last theorem, the modularity theorem was the main active area of research for which the proof was developed, and not just a historical oddity, so the time spent on its work could be justified from a professional point of view. However, the general consensus was that solving the Taniyama-Shimura hypothesis proved to be inexpedient.
Fermat's Last Theorem: Wiles' proof
Upon learning that Ribet had proven Frey's theory correct, English mathematician Andrew Wiles, who had been interested in Fermat's Last Theorem since childhood and had experience with elliptic curves and adjacent domains, decided to try to prove the Taniyama-Shimura Conjecture as a way to prove Fermat's Last Theorem. In 1993, six years after announcing his goal, while secretly working on the problem of solving the theorem, Wiles managed to prove a related conjecture, which in turn would help him prove Fermat's last theorem. Wiles' document was enormous in size and scope.
A flaw was discovered in one part of his original paper during peer review and required another year of collaboration with Richard Taylor to jointly solve the theorem. As a result, Wiles' final proof of Fermat's Last Theorem was not long in coming. In 1995, it was published on a much smaller scale than Wiles's previous mathematical work, illustrating that he was not mistaken in his previous conclusions about the possibility of proving the theorem. Wiles' achievement was widely publicized in the popular press and popularized in books and television programs. The remaining parts of the Taniyama-Shimura-Weyl conjecture, which have now been proven and are known as the modularity theorem, were subsequently proved by other mathematicians who built on Wiles' work between 1996 and 2001. For his achievement, Wiles has been honored and received numerous awards, including the 2016 Abel Prize.
Wiles' proof of Fermat's last theorem is a special case of solving the modularity theorem for elliptic curves. However, this is the most famous case of such a large-scale mathematical operation. Along with solving Ribe's theorem, the British mathematician also obtained a proof of Fermat's last theorem. Fermat's Last Theorem and Modularity Theorem were almost universally considered unprovable by modern mathematicians, but Andrew Wiles was able to prove to the scientific world that even pundits can be wrong.
Wiles first announced his discovery on Wednesday 23 June 1993 at a Cambridge lecture titled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993, it was found that his calculations contained an error. A year later, on September 19, 1994, in what he would call "the most important moment of his working life," Wiles stumbled upon a revelation that allowed him to correct the problem solution to the point where it could satisfy the mathematical community.
Job Description
Andrew Wiles' proof of Fermat's Theorem uses many methods from algebraic geometry and number theory, and has many ramifications in these areas of mathematics. He also uses the standard constructions of modern algebraic geometry, such as the category of schemes and the Iwasawa theory, as well as other 20th-century methods that were not available to Pierre de Fermat.
The two papers containing the evidence are 129 pages long and were written over the course of seven years. John Coates described this discovery as one of greatest achievements number theory, and John Conway called it the main mathematical achievement of the 20th century. Wiles, in order to prove Fermat's last theorem by proving the modularity theorem for the special case of semistable elliptic curves, developed powerful methods for lifting modularity and opened up new approaches to numerous other problems. For solving Fermat's last theorem, he was knighted and received other awards. When it became known that Wiles had won the Abel Prize, the Norwegian Academy of Sciences described his achievement as "a delightful and elementary proof of Fermat's Last Theorem".
How it was
One of the people who reviewed Wiles' original manuscript with the solution to the theorem was Nick Katz. In the course of his review, he asked the Briton a number of clarifying questions that led Wiles to admit that his work clearly contains a gap. In one critical part of the proof, an error was made that gave an estimate for the order of a particular group: the Euler system used to extend the Kolyvagin and Flach method was incomplete. The mistake, however, did not make his work useless - every part of Wiles's work was very significant and innovative in itself, as were many of the developments and methods that he created in the course of his work and which affected only one part of the manuscript. However, this original work, published in 1993, did not really have a proof of Fermat's Last Theorem.
Wiles spent almost a year trying to rediscover a solution to the theorem, first alone and then in collaboration with his former student Richard Taylor, but all seemed to be in vain. By the end of 1993, rumors had circulated that Wiles's proof had failed in testing, but how serious this failure was was not known. Mathematicians began to put pressure on Wiles to reveal the details of his work, whether it was done or not, so that the wider community of mathematicians could explore and use whatever he was able to achieve. Instead of quickly correcting his mistake, Wiles only discovered additional difficult aspects in the proof of Fermat's Last Theorem, and finally realized how difficult it was.
Wiles states that on the morning of September 19, 1994, he was on the verge of giving up and giving up, and was almost resigned to failing. He was ready to publish his unfinished work so that others could build on it and find where he was wrong. The English mathematician decided to give himself one last chance and analyzed the theorem for the last time to try to understand the main reasons why his approach did not work, when he suddenly realized that the Kolyvagin-Flak approach would not work until he connected more and more to the proof process Iwasawa's theory by making it work.
On October 6, Wiles asked three colleagues (including Fultins) to review him new job, and on October 24, 1994, he submitted two manuscripts - "Modular elliptic curves and Fermat's last theorem" and "Theoretical properties of the ring of some Hecke algebras", the second of which Wiles wrote jointly with Taylor and proved that certain conditions necessary for to justify the corrected step in the main article.
These two papers were reviewed and finally published as a full text edition in the May 1995 Annals of Mathematics. Andrew's new calculations were widely analyzed and eventually accepted by the scientific community. In these works, the modularity theorem for semistable elliptic curves was established - the last step towards proving Fermat's Last Theorem, 358 years after it was created.
History of the Great Problem
Solving this theorem has been considered the biggest problem in mathematics for many centuries. In 1816 and in 1850 the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the Academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he did not apply for the prize. Another prize was offered to him in 1883 by the Brussels Academy.
Wolfskel Prize
In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks (a large amount for the time) to the Göttingen Academy of Sciences to be the prize for the complete proof of Fermat's Last Theorem. On June 27, 1908, the Academy published nine award rules. Among other things, these rules required the proof to be published in a peer-reviewed journal. The prize was to be awarded only two years after publication. The competition was due to expire on September 13, 2007 - about a century after it began. On June 27, 1997, Wiles received Wolfschel's prize money and then another $50,000. In March 2016, he received €600,000 from the Norwegian government as part of the Abel Prize for "an amazing proof of Fermat's last theorem with the help of the modularity conjecture for semistable elliptic curves, opening a new era in number theory." It was the world triumph of the humble Englishman.
Prior to Wiles's proof, Fermat's Theorem, as mentioned earlier, was considered absolutely unsolvable for centuries. Thousands of incorrect evidence at various times were presented to the Wolfskell committee, amounting to approximately 10 feet (3 meters) of correspondence. Only in the first year of the existence of the prize (1907-1908) 621 applications were submitted claiming to solve the theorem, although by the 1970s their number had decreased to about 3-4 applications per month. According to F. Schlichting, Wolfschel's reviewer, most of the evidence was based on elementary methods taught in schools and was often presented as "people with a technical background but a failed career". According to the historian of mathematics Howard Aves, Fermat's last theorem set a kind of record - it is the theorem with the most incorrect proofs.
Fermat's laurels went to the Japanese
As discussed earlier, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama discovered a possible connection between two apparently completely different branches of mathematics - elliptic curves and modular forms. The resulting modularity theorem (then known as the Taniyama-Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.
The theory was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist André Weil found evidence to support the Japanese conclusions. As a result, the hypothesis has often been referred to as the Taniyama-Shimura-Weil hypothesis. It became part of the Langlands program, which is a list of important hypotheses that need to be proven in the future.
Even after serious scrutiny, the conjecture has been recognized by modern mathematicians as extremely difficult, or perhaps inaccessible to proof. Now it is this theorem that is waiting for its Andrew Wiles, who could surprise the whole world with its solution.
Fermat's Theorem: Perelman's proof
Despite the common myth, the Russian mathematician Grigory Perelman, for all his genius, has nothing to do with Fermat's theorem. That, however, does not detract from his numerous merits to the scientific community.