Fermat's Last Theorem: Wiles and Perelman's proof, formulas, calculation rules and complete proof of the theorem. The sensation around Fermat's theorem turned out to be a misunderstanding Fermat's Last Theorem: Wiles' proof
Andrew Wiles is a professor of mathematics at Princeton University, he proved Fermat's Last Theorem, which generations of scientists have struggled with for hundreds of years.
30 years on one task
Wiles first learned about Fermat's last theorem when he was ten years old. He stopped by the library on his way home from school and became engrossed in reading the book “The Final Problem” by Eric Temple Bell. Perhaps without even knowing it, from that moment on he dedicated his life to the search for proof, despite the fact that it was something that had eluded the best minds on the planet for three centuries.
Wiles learned about Fermat's last theorem when he was ten years old
He found it 30 years later after another scientist, Ken Ribet, proved the connection between the theorem of Japanese mathematicians Taniyama and Shimura with Fermat's Last Theorem. Unlike his skeptical colleagues, Wiles immediately understood that this was it, and seven years later he put an end to the proof.
The process of proof itself turned out to be very dramatic: Wiles completed his work in 1993, but right during his public appearance he found a significant “gap” in his reasoning. It took two months to find an error in the calculations (the error was hidden among 130 printed pages of the solution to the equation). Then, for a year and a half, intense work was carried out to correct the error. The entire scientific community of the Earth was at a loss. Wiles completed his work on September 19, 1994 and immediately presented it to the public.
Frightening Glory
Andrew's greatest fear was fame and publicity. He refused to appear on television for a very long time. It is believed that John Lynch was able to convince him. He assured Wiles that he could inspire a new generation of mathematicians and show the power of mathematics to the public.
Andrew Wiles refused to appear on television for a long time
A little later, a grateful society began to reward Andrew with prizes. So on June 27, 1997, Wiles received the Wolfskehl Prize, which amounted to approximately $50,000. This is much less than Wolfskehl intended to leave a century earlier, but hyperinflation led to a reduction in the amount.
Unfortunately, the mathematical equivalent of the Nobel Prize, the Fields Prize, simply did not go to Wiles due to the fact that it is awarded to mathematicians under forty years of age. Instead, he received a special silver plate at the Fields Medal ceremony in honor of his important achievement. Wiles has also won the prestigious Wolf Prize, the King Faisal Prize and many other international awards.
Colleagues' opinions
The reaction of one of the most famous modern Russian mathematicians, Academician V. I. Arnold, to the proof is “actively skeptical”:
This is not real mathematics - real mathematics is geometric and has strong connections with physics. Moreover, Fermat's problem itself, by its nature, cannot generate the development of mathematics, since it is “binary”, that is, the formulation of the problem requires an answer only to the “yes or no” question.
At the same time, the mathematical works of V. I. Arnold himself in recent years turned out to be largely devoted to variations on very similar number-theoretic topics. It is possible that Wiles, paradoxically, became an indirect cause of this activity.
A real dream
When Andrew is asked how he managed to sit within four walls for more than 7 years doing one task, Wiles tells how he dreamed during his work thatThe time will come when mathematics courses in universities, and even in schools, will be adjusted to his method of proving the theorem. He wanted the proof of Fermat's Last Theorem to become not only a model mathematical problem, but also a methodological model for teaching mathematics. Wiles imagined that using her example it would be possible to study all the main branches of mathematics and physics.
4 ladies without whom there would be no proof
Andrew is married and has three daughters, two of whom were born "during the seven-year process of the first draft of the proof."
Wiles himself believes that without his family he would not have succeeded.
During these years, only Nada, Andrew's wife, knew that he was storming alone the most inaccessible and most famous peak of mathematics. It is to them, Nadya, Claire, Kate and Olivia, that Wiles’s famous final article “Modular elliptic curves and Fermat’s Last Theorem” in the central mathematical journal “Annals of Mathematics” is dedicated, where the most important mathematical works are published. However, Wiles himself does not deny at all that without his family he would not have succeeded.
Mathematician Andrew Wiles received the Abel Prize for his proof of Fermat's theorem
An honorary award, called the "Nobel Prize for mathematicians", was awarded to him for his proof of Fermat's Last Theorem in 1994
Andrew Wiles
© AP Photo/Charles Rex Arbogast, archive
OSLO, March 15. /Corr. TASS Yuri Mikhailenko/. Briton Andrew Wiles has been announced as the winner of the Abel Prize, awarded by the Norwegian Academy of Sciences. The honorary award, often called the “Nobel Prize for mathematicians,” was awarded to him for his proof of Fermat’s Last Theorem in 1994, which “launched a new era in number theory.”
“The new ideas introduced by Wiles opened up the possibility of further breakthroughs,” said Abel Committee Chairman Jon Rognes. “Few mathematical problems have as rich a scientific history and as spectacular a proof as Fermat’s Last Theorem.”
Sir Andrew's scientific journey
In comments to the Norwegian Telegraph Bureau, Rognes also clarified that the proof of the famous theorem was just one of the reasons why Wiles was chosen among the candidates nominated for the prize this year.
“To solve a theorem that could not be proven for 350 years, he used the approaches of two modern branches of mathematical science, studying, in particular, semi-stable elliptic curves,” Rognes told reporters. “Such mathematics is used, for example, in elliptic cryptography, with the help of which security data on payments made using plastic cards."
The scientist, who turns 63 next month, was educated at Oxford and Cambridge universities. His father was an Anglican clergyman and was professor of theology at Cambridge for more than 20 years. Wiles himself worked in the United States for 30 years, teaching at Princeton University, and from 2005 to 2009 headed the mathematics department there. He currently works in Oxford. He has won a dozen mathematical prizes, and for his scientific achievements he was also knighted by Queen Elizabeth II of Great Britain.
Deceptive simplicity
The peculiarity of the theorem, formulated by the Frenchman Pierre Fermat (1601 - 1665), is in a deceptively simple formulation: the equation “A to the power of n plus B to the power of n is equal to C to the power of n” has no natural solutions if the number n is greater than two. At first glance, it suggests a fairly simple proof, but in reality this turns out to be completely different.
Wiles himself admitted in numerous interviews that the theorem intrigued him at the age of 10. Even then, it was easy for him to understand the conditions of the problem, and he was haunted by the fact that for three centuries not a single mathematician had been able to solve it. The childhood hobby has not faded over the years. Having already made a scientific career, Wiles spent many years struggling with the solution in his free time, but did not advertise it, since among his colleagues, a passion for Fermat’s theorem was considered bad manners. He proposed his proof, based on the hypothesis of two Japanese scientists, and published it in 1993, but a few months later an error was discovered in his calculations.
For more than a year, Wiles, together with his students, tried to correct it, almost giving up in the end, but ultimately still found a proof that was recognized as correct. At the same time, the supposedly existing simple and elegant proof, which Fermat himself mentioned, has not yet been found.
Who was Henrik Abel
In 2014 and 2009, the Abel Prize laureates were students of the Russian mathematical school - Yakov Sinai and Mikhail Gromov, respectively. The award is named after the famous Norwegian Niels Henrik Abel. He became the founder of the theory of elliptic functions and made significant contributions to the theory of series.
In honor of the 200th anniversary of the birth of the scientist, who lived only 26 years, the Norwegian government in 2002 allocated 200 million kroner (about $23.4 million at current exchange rates) to establish the Abel Foundation and the Abel Prize. It is intended not only to celebrate the merits of outstanding mathematicians, but also to contribute to the growing popularity of this scientific discipline among young people.
Today, the cash component of the award is 6 million crowns ($700 thousand). The official award ceremony is scheduled to take place on May 24. The honorary award will be presented to the laureate by the heir to the Norwegian throne, Prince Haakon Magnus.
There are not many people in the world who have never heard of Fermat’s Last Theorem - perhaps this is the only mathematical problem that has become so widely known and has become a real legend. It is mentioned in many books and films, and the main context of almost all mentions is the impossibility of proving the theorem.
Yes, this theorem is very well known and, in a sense, has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this happened back in 1995. But first things first.
So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in essence and understandable to anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.
Why is she so famous? Now we'll find out...
Are there many proven, unproven and as yet unproven theorems? The point here is that Fermat's Last Theorem represents the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult problem, and yet its formulation can be understood by anyone with the 5th grade of high school, but not even every professional mathematician can understand the proof. Neither in physics, nor in chemistry, nor in biology, nor in mathematics, is there a single problem that could be formulated so simply, but remained unsolved for so long. 2. What does it consist of?
Let's start with Pythagorean pants. The wording is really simple - at first glance. As we know from childhood, “Pythagorean pants are equal on all sides.” The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs.
In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triplets satisfying the equality x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They probably tried to look for C's and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the brotherhood were more philosophers and aesthetes than mathematicians.
That is, it is easy to select a set of numbers that perfectly satisfy the equality x²+y²=z²
Starting from 3, 4, 5 - indeed, a junior student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
So, it turns out that they are NOT. This is where the trick begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, its absence. When you need to prove that there is a solution, you can and should simply present this solution.
Proving absence is more difficult: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give solution). And that’s it, the opponent is defeated. How to prove absence?
Say: “I haven’t found such solutions”? Or maybe you weren't looking well? What if they exist, only very large, very large, such that even a super-powerful computer still doesn’t have enough strength? This is what’s difficult.
This can be shown visually like this: if you take two squares of suitable sizes and disassemble them into unit squares, then from this bunch of unit squares you get a third square (Fig. 2): 
But let’s do the same with the third dimension (Fig. 3) - it doesn’t work. There are not enough cubes, or there are extra ones left:
But the 17th century mathematician Frenchman Pierre de Fermat enthusiastically studied the general equation x n + y n = z n. And finally, I concluded: for n>2 there are no integer solutions. Fermat's proof is irretrievably lost. Manuscripts are burning! All that remains is his remark in Diophantus’ Arithmetic: “I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it.”
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never making mistakes. Even if he did not leave evidence of a statement, it was subsequently confirmed. Moreover, Fermat proved his thesis for n=4. Thus, the hypothesis of the French mathematician went down in history as Fermat’s Last Theorem.
After Fermat, such great minds as Leonhard Euler worked on the search for a proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat’s Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat’s last theorem was practically over.
It is easily shown that it is enough to prove Fermat’s theorem only for simple n: 3, 5, 7, 11, 13, 17, ... For composite n, the proof remains valid. But there are infinitely many prime numbers...
In 1825, using the method of Sophie Germain, female mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, using the same method, the Frenchman Gabriel Lame showed the truth of the theorem for n=7. Gradually the theorem was proven for almost all n less than one hundred.
Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that the theorem in general cannot be proven using the methods of mathematics of the 19th century. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unawarded.
In 1907, the wealthy German industrialist Paul Wolfskehl decided to take his own life because of unrequited love. Like a true German, he set the date and time of suicide: exactly at midnight. On the last day he made a will and wrote letters to friends and relatives. Things ended before midnight. It must be said that Paul was interested in mathematics. Having nothing else to do, he went to the library and began to read Kummer's famous article. Suddenly it seemed to him that Kummer had made a mistake in his reasoning. Wolfskel began to analyze this part of the article with a pencil in his hands. Midnight has passed, morning has come. The gap in the proof has been filled. And the very reason for suicide now looked completely ridiculous. Paul tore up his farewell letters and rewrote his will.
He soon died of natural causes. The heirs were quite surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskehl Prize. 100,000 marks were awarded to the person who proved Fermat's theorem. Not a pfennig was awarded for refuting the theorem...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and resolutely refused to waste time on such a useless exercise. But the amateurs had a blast. A few weeks after the announcement, an avalanche of “evidence” hit the University of Göttingen. Professor E.M. Landau, whose responsibility was to analyze the evidence sent, distributed cards to his students:
Dear. . . . . . . .
Thank you for sending me the manuscript with the proof of Fermat’s Last Theorem. The first error is on page ... in line... . Because of it, the entire proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, relying on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems - the continuum hypothesis. What if Fermat's Last Theorem is also undecidable?! But true Great Theorem fanatics were not disappointed at all. The advent of computers suddenly gave mathematicians a new method of proof. After World War II, teams of programmers and mathematicians proved Fermat's Last Theorem for all values of n up to 500, then up to 1,000, and later up to 10,000.
In the 1980s, Samuel Wagstaff raised the limit to 25,000, and in the 1990s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if you subtract even a trillion trillion from infinity, it will not become smaller. Mathematicians are not convinced by statistics. To prove the Great Theorem meant to prove it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends began researching modular forms. These forms generate series of numbers, each with its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. No connection has ever been found between such different objects.
However, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of an entire direction in mathematics, but until the Taniyama-Shimura hypothesis was proven, the entire building could collapse at any moment.
In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation could not have a counterpart in the modular world. From now on, Fermat's Last Theorem was inextricably linked with the Taniyama-Shimura conjecture. Having proven that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proven. But for thirty years it was not possible to prove the Taniyama-Shimura hypothesis, and there was less and less hope for success.
In 1963, when he was just ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not give up on it. As a schoolboy, student, and graduate student, he prepared himself for this task.
Having learned about Ken Ribet's findings, Wiles plunged headlong into proving the Taniyama-Shimura hypothesis. He decided to work in complete isolation and secrecy. “I realized that everything that had anything to do with Fermat’s Last Theorem arouses too much interest... Too many spectators obviously interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.
In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational paper at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.
While the hype continued in the press, serious work began to verify the evidence. Every piece of evidence must be carefully examined before the evidence can be considered rigorous and accurate. Wiles spent a restless summer waiting for feedback from reviewers, hoping that he would be able to win their approval. At the end of August, experts found the judgment to be insufficiently substantiated.
It turned out that this decision contains a gross error, although in general it is correct. Wiles did not give up, called on the help of the famous specialist in number theory Richard Taylor, and already in 1994 they published a corrected and expanded proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the mathematical journal “Annals of Mathematics”. But the story did not end there either - the final point was reached only in the next year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.
“...half a minute after the start of the festive dinner on the occasion of her birthday, I presented Nadya with the manuscript of the complete proof” (Andrew Wales). Have I not yet said that mathematicians are strange people?
This time there was no doubt about the evidence. Two articles were subjected to the most careful analysis and were published in May 1995 in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society that Fermat’s Last Theorem is unsolvable. But even those who know about the proof found continue to work in this direction - few are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the efforts of many mathematicians (mostly amateurs, not professional scientists) are thrown into the search for a simple and concise proof, but this path, most likely, will not lead anywhere...
source
So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in nature and understandable to anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.
Why is she so famous? Now we'll find out...
Are there many proven, unproven and as yet unproven theorems? The point here is that Fermat's Last Theorem represents the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult problem, and yet its formulation can be understood by anyone with the 5th grade of high school, but not even every professional mathematician can understand the proof. Neither in physics, nor in chemistry, nor in biology, nor in mathematics, is there a single problem that could be formulated so simply, but remained unsolved for so long. 2. What does it consist of?
Let's start with Pythagorean pants. The wording is really simple - at first glance. As we know from childhood, “Pythagorean pants are equal on all sides.” The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs.
In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triplets satisfying the equality x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They probably tried to look for C's and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the brotherhood were more philosophers and aesthetes than mathematicians.
That is, it is easy to select a set of numbers that perfectly satisfy the equality x²+y²=z²
Starting from 3, 4, 5 - indeed, a junior student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
And so on. What if we take a similar equation x³+y³=z³? Maybe there are such numbers too?
And so on (Fig. 1).
So, it turns out that they are NOT. This is where the trick begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, its absence. When you need to prove that there is a solution, you can and should simply present this solution.
Proving absence is more difficult: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give solution). And that’s it, the opponent is defeated. How to prove absence?
Say: “I haven’t found such solutions”? Or maybe you weren't looking well? What if they exist, only very large, very large, such that even a super-powerful computer still doesn’t have enough strength? This is what’s difficult.
This can be shown visually like this: if you take two squares of suitable sizes and disassemble them into unit squares, then from this bunch of unit squares you get a third square (Fig. 2):
But let’s do the same with the third dimension (Fig. 3) – it doesn’t work. There are not enough cubes, or there are extra ones left:
But the 17th century French mathematician Pierre de Fermat enthusiastically studied the general equation x n +y n =z n . And finally, I concluded: for n>2 there are no integer solutions. Fermat's proof is irretrievably lost. Manuscripts are burning! All that remains is his remark in Diophantus’ Arithmetic: “I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it.”
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never making mistakes. Even if he did not leave evidence of a statement, it was subsequently confirmed. Moreover, Fermat proved his thesis for n=4. Thus, the hypothesis of the French mathematician went down in history as Fermat’s Last Theorem.
After Fermat, such great minds as Leonhard Euler worked on the search for a proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat’s Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat’s last theorem was practically over.
It is easily shown that it is enough to prove Fermat’s theorem only for simple n: 3, 5, 7, 11, 13, 17, ... For composite n, the proof remains valid. But there are infinitely many prime numbers...
In 1825, using the method of Sophie Germain, female mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, using the same method, the Frenchman Gabriel Lame showed the truth of the theorem for n=7. Gradually the theorem was proven for almost all n less than one hundred.
Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that the theorem in general cannot be proven using the methods of mathematics of the 19th century. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unawarded.
In 1907, the wealthy German industrialist Paul Wolfskehl decided to take his own life because of unrequited love. Like a true German, he set the date and time of suicide: exactly at midnight. On the last day he made a will and wrote letters to friends and relatives. Things ended before midnight. It must be said that Paul was interested in mathematics. Having nothing else to do, he went to the library and began to read Kummer's famous article. Suddenly it seemed to him that Kummer had made a mistake in his reasoning. Wolfskel began to analyze this part of the article with a pencil in his hands. Midnight has passed, morning has come. The gap in the proof has been filled. And the very reason for suicide now looked completely ridiculous. Paul tore up his farewell letters and rewrote his will.
He soon died of natural causes. The heirs were quite surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskehl Prize. 100,000 marks were awarded to the person who proved Fermat's theorem. Not a pfennig was awarded for refuting the theorem...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and resolutely refused to waste time on such a useless exercise. But the amateurs had a blast. A few weeks after the announcement, an avalanche of “evidence” hit the University of Göttingen. Professor E.M. Landau, whose responsibility was to analyze the evidence sent, distributed cards to his students:
Dear. . . . . . . .
Thank you for sending me the manuscript with the proof of Fermat’s Last Theorem. The first error is on page ... in line... . Because of it, the entire proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, relying on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems - the continuum hypothesis. What if Fermat's Last Theorem is also undecidable?! But true Great Theorem fanatics were not disappointed at all. The advent of computers suddenly gave mathematicians a new method of proof. After World War II, teams of programmers and mathematicians proved Fermat's Last Theorem for all values of n up to 500, then up to 1,000, and later up to 10,000.
In the 1980s, Samuel Wagstaff raised the limit to 25,000, and in the 1990s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if you subtract even a trillion trillion from infinity, it will not become smaller. Mathematicians are not convinced by statistics. To prove the Great Theorem meant to prove it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends began researching modular forms. These forms generate series of numbers, each with its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. No connection has ever been found between such different objects.
However, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of an entire direction in mathematics, but until the Taniyama-Shimura hypothesis was proven, the entire building could collapse at any moment.
In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation could not have a counterpart in the modular world. From now on, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proven that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proven. But for thirty years it was not possible to prove the Taniyama-Shimura hypothesis, and there was less and less hope for success.
In 1963, when he was just ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not give up on it. As a schoolboy, student, and graduate student, he prepared himself for this task.
Having learned about Ken Ribet's findings, Wiles plunged headlong into proving the Taniyama-Shimura conjecture. 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...
