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... August 5th, 2013
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
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.
Judging by the popularity of the query "Fermat's theorem - short proof" this mathematical problem really interests many people. This theorem was first stated by Pierre de Fermat in 1637 on the edge of a copy of 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, a complete proof of Fermat's theorem by Andrew Wiles. It was described as "stunning progress" and led Wiles to receive the Abel Prize in 2016. While 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 for raising modularity. These achievements advanced mathematics by 100 years. The proof of Fermat's little theorem is not something out of the ordinary today.
The unsolved 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. It is one of the most notable theorems in the history of mathematics and, prior to the complete proof of Fermat's last theorem by division, it was in the Guinness Book of Records as the "hardest mathematical problem", one of the features of which is that it has the largest number of failed proofs.
Historical 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 margin of a book that the more general equation a n + b n = c n had no solutions in natural numbers if n was an integer greater than 2. Although Fermat himself claimed to have a solution to his problem, he did not leave no details about her proof. The elementary proof of Fermat's theorem, stated 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 sparked significant developments in number theory, and over time, Fermat's Last Theorem became known as an unsolved problem in mathematics.
Brief history of evidence
If n = 4, as Fermat himself proved, it is enough to prove the theorem for indices n, which are prime numbers. Over the next two centuries (1637-1839) the conjecture was proven only for the prime numbers 3, 5 and 7, although Sophie Germain updated and proved an approach that applied to the entire class of prime numbers. In the mid-19th century, Ernst Kummer expanded on this and proved the theorem for all regular primes, causing irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer research, other mathematicians were able to expand the solution to the theorem, aiming to cover all major exponents up to four million, but the proof for all exponents was still unavailable (meaning that mathematicians generally considered the solution to the theorem impossible, extremely difficult, or unattainable with current knowledge).
Work by 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 completely different areas of mathematics. Known at the time as the Taniyama-Shimura-Weil conjecture and (eventually) as the modularity theorem, it stood on its own, with no apparent connection to Fermat's last theorem. It was widely regarded as an important mathematical theorem in its own right, but was considered (like Fermat's theorem) impossible to prove. At the same time, the proof of Fermat's great theorem (by the method of division and the use of complex mathematical formulas) was carried out only half a century later.
In 1984, Gerhard Frey noticed an obvious connection between these two previously unrelated and unresolved problems. Complete proof that the two theorems were closely related was published in 1986 by Ken Ribet, who built on a partial proof by Jean-Pierre Serres, who proved all but one part, known as the "epsilon conjecture". Simply put, these works by Frey, Serres and Ribe showed that if the modularity theorem could be proven for at least a semistable class of elliptic curves, then the proof of Fermat's last theorem would also be discovered sooner or later. 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 it should have been proven soon.
Although both theorems were difficult 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 proven for all numbers, not just some. Important to the researchers who chose the research topic was the fact that, unlike Fermat's last theorem, the modularity theorem was a major active area of research for which a proof had been developed, and not just a historical oddity, so the time spent working on it could be justified from a professional point of view. However, the general consensus was that solving the Taniyama-Shimura conjecture was not practical.
Fermat's Last Theorem: Wiles' Proof
After 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 working with elliptic curves and related fields, 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.
The 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, clearly showing that he was not mistaken in his previous conclusions about the possibility of proving the theorem. Wiles' achievement was widely reported in the popular press and popularized in books and television programs. The remaining parts of the Taniyama-Shimura-Weil conjecture, which have now been proven and are known as the modularity theorem, were subsequently proven by other mathematicians who built on Wiles' work between 1996 and 2001. For his achievement, Wiles was honored and received numerous awards, including the 2016 Abel Prize.
Wiles's proof of Fermat's last theorem is a special case of a solution to the modularity theorem for elliptic curves. However, this is the most famous case of such a large-scale mathematical operation. Along with solving Ribet's theorem, the British mathematician also obtained a proof of Fermat's last theorem. Fermat's Last Theorem and the Modularity Theorem were almost universally considered unprovable by modern mathematicians, but Andrew Wiles was able to prove to the entire scientific world that even pundits can be mistaken.
Wiles first announced his discovery on Wednesday 23 June 1993 in a lecture at Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993 it was determined 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 solution to the problem to the point where it could satisfy the mathematical community.
Characteristics of work
Andrew Wiles's proof of Fermat's theorem uses many techniques from algebraic geometry and number theory and has many ramifications in these areas of mathematics. He also uses standard constructs of modern algebraic geometry, such as the category of schemes and Iwasawa theory, as well as other 20th-century methods that were not available to Pierre Fermat.
The two articles containing the evidence total 129 pages and were written over seven years. John Coates described this discovery as one of the greatest achievements of 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 discovered new approaches to numerous other problems. For solving Fermat's last theorem he was knighted and received other awards. When it was announced that Wiles had won the Abel Prize, the Norwegian Academy of Sciences described his achievement as "a marvelous and elementary proof of Fermat's last theorem."
How it was
One of the people who analyzed Wiles' original manuscript of the theorem's solution was Nick Katz. During his review, he asked the Briton a series of clarifying questions, which forced Wiles to admit that his work clearly contained a gap. There was an error in one critical part of the proof 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 render his work useless - each part of Wiles' 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 actually provide a proof of Fermat's Last Theorem.
Wiles spent almost a year trying to rediscover the 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 spread that Wiles' proof had failed in testing, but how serious the failure was was not known. Mathematicians began to put pressure on Wiles to reveal the details of his work, whether it was completed or not, so that the wider community of mathematicians could explore and use everything he had achieved. Instead of quickly correcting his mistake, Wiles only discovered additional complexities 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 almost resigned himself to the fact that he had failed. He was willing to publish his unfinished work so that others could build on it and find where he had gone wrong. The English mathematician decided to give himself one last chance and analyzed the theorem one last time to try to understand the main reasons why his approach did not work, when he suddenly realized that the Kolyvagin-Flac approach would not work until he also included proof in the process Iwasawa's theory, making it work.
On October 6, Wiles asked three colleagues (including Faltins) to review his new work, 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 co-wrote with Taylor and argued that certain conditions necessary to justify the corrected step in the main article were met.
These two papers were reviewed and finally published as a full-text edition in the May 1995 issue of the Annals of Mathematics. Andrew's new calculations were widely analyzed and eventually accepted by the scientific community. These works established the modularity theorem for semistable elliptic curves, the final 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 again 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 into ideal numbers, although he did not apply for the prize. Another prize was offered to him in 1883 by the Brussels Academy.
Wolfskehl Prize
In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks (a large sum for that time) to the Göttingen Academy of Sciences as a prize for a complete proof of Fermat's last theorem. On June 27, 1908, the Academy published nine awards rules. Among other things, these rules required publication of the evidence in a peer-reviewed journal. The prize was not to be awarded until two years after publication. The competition was due to expire on September 13, 2007 - approximately 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 his "stunning proof of Fermat's last theorem using the modularity conjecture for semistable elliptic curves, opening a new era in number theory." It was a world triumph for the humble Englishman.
Before Wiles' proof, Fermat's theorem, as mentioned earlier, was considered absolutely unsolvable for centuries. Thousands of incorrect evidence were presented to Wolfskehl's committee at various times, amounting to approximately 10 feet (3 meters) of correspondence. In the first year of the prize's existence alone (1907-1908), 621 applications were submitted claiming to solve the theorem, although by the 1970s this number had decreased to approximately 3-4 applications per month. According to F. Schlichting, Wolfschel's reviewer, most of the evidence was based on rudimentary methods taught in schools and was often presented by "people with a technical background but an unsuccessful 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 laurels went to the Japanese
As mentioned 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) from their research 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 Andre Weyl found evidence to support the Japanese's findings. As a result, the conjecture was often called the Taniyama-Shimura-Weil conjecture. It became part of the Langlands program, which is a list of important hypotheses that require proof in the future.
Even after serious attention, the conjecture was recognized by modern mathematicians as extremely difficult or perhaps impossible to prove. Now it is this theorem that is waiting for Andrew Wiles, who could surprise the whole world with its solution.
Fermat's theorem: Perelman's proof
Despite the popular myth, the Russian mathematician Grigory Perelman, for all his genius, has nothing to do with Fermat’s theorem. Which, however, does not in any way detract from his numerous services to the scientific community.
In the last twentieth century, an event occurred that has never been equal in scale in mathematics in its entire history. On September 19, 1994, a theorem formulated by Pierre de Fermat (1601-1665) more than 350 years ago in 1637 was proven. It is also known as "Fermat's last theorem" or "Fermat's last theorem" because there is also the so-called "Fermat's little theorem". It was proved by 41-year-old Princeton University professor Andrew Wiles, who up to this point had been unremarkable in the mathematical community and, by mathematical standards, no longer young.
It is surprising that not only our ordinary Russian inhabitants, but also many people interested in science, including even a considerable number of scientists in Russia who use mathematics in one way or another, do not really know about this event. This is shown by the continuous “sensational” reports about “elementary proofs” of Fermat’s theorem in Russian popular newspapers and on television. The latest evidence was covered with such informational power, as if Wiles’ evidence, which had undergone the most authoritative examination and became widely known throughout the world, did not exist. The reaction of the Russian mathematical community to this front-page news in the context of a rigorous proof obtained long ago was surprisingly sluggish. Our goal is to sketch the fascinating and dramatic history of Wiles's proof in the context of the enchanting history of Fermat's great theorem itself, and to talk a little about its proof itself. Here we are primarily interested in the question of the possibility of an accessible presentation of Wiles’ proof, which, of course, most mathematicians in the world know about, but only very, very few of them can talk about understanding this proof.
So, let's remember Fermat's famous theorem. Most of us have heard about it in one way or another since school. This theorem is related to a very significant equation. This is perhaps the simplest meaningful equation that can be written using three unknowns and one more strictly positive integer parameter. Here it is:
Fermat's Last Theorem states that for values of the parameter (the degree of the equation) greater than two, there are no integer solutions to a given equation (except, of course, the solution when all these variables are equal to zero at the same time).
The attractive power of Fermat’s theorem for the general public is obvious: there is no other mathematical statement that has such simplicity of formulation, the apparent accessibility of the proof, as well as the attractiveness of its “status” in the eyes of society.
Before Wiles, an additional incentive for the Fermatists (as people who maniacally attacked Fermat’s problem were called) was the German Wolfskehl’s prize for proof, established almost a hundred years ago, although small compared to the Nobel Prize - it managed to depreciate during the First World War.
In addition, the probable elementary nature of the proof has always attracted attention, since Fermat himself “proved it” by writing in the margins of the translation of Diophantus’ Arithmetic: “I have found a truly wonderful proof of this, but the margins here are too narrow to contain it.”
That is why it is appropriate here to give an assessment of the relevance of popularizing Wiles’ proof of Fermat’s problem, which belongs to the famous American mathematician R. Murty (we quote from the soon-to-be-released translation of the book by Yu. Manin and A. Panchishkin “Introduction to Modern Number Theory”):
“Fermat’s Last Theorem occupies a special place in the history of civilization. With its outward simplicity, it has always attracted both amateurs and professionals... Everything looks as if it was conceived by some higher mind, which over the centuries developed various lines of thought only to then reunite them into one exciting fusion to solve the Great Fermat's theorems. No one person can claim to be an expert on all the ideas used in this “miracle” proof. In an era of universal specialization, when each of us knows “more and more about less and less,” it is absolutely necessary to have an overview of this masterpiece...”
Let's start with a brief historical excursion, mainly inspired by Simon Singh's fascinating book Fermat's Last Theorem. Serious passions have always been boiling around the insidious theorem, alluring with its apparent simplicity. The history of its proof is full of drama, mysticism and even direct victims. Perhaps the most iconic victim is Yutaka Taniyama (1927-1958). It was this young talented Japanese mathematician, distinguished by great extravagance in life, who created the basis for Wiles' attack in 1955. Based on his ideas, Goro Shimura and Andre Weil a few years later (60-67) finally formulated the famous conjecture, having proved a significant part of which, Wiles obtained Fermat’s theorem as a corollary. The mysticism of the death story of the non-trivial Yutaka is associated with his stormy temperament: he hanged himself at the age of thirty-one due to unhappy love.
The entire long history of the mysterious theorem was accompanied by constant announcements about its proof, starting with Fermat himself. Constant errors in the endless stream of proofs befell not only amateur mathematicians, but also professional mathematicians. This led to the fact that the term "Fermatist", applied to those who proved Fermat's theorem, became a common noun. The constant intrigue with its proof sometimes led to funny incidents. So, when a gap was discovered in the first version of Wiles’ already widely publicized proof, a malicious inscription appeared at one of the New York subway stations: “I have found a truly wonderful proof of Fermat’s Last Theorem, but my train has arrived and I don’t have time to write it down.”
Andrew Wiles, born in England in 1953, studied mathematics at Cambridge; in graduate school he studied with Professor John Coates. Under his guidance, Andrew comprehended the theory of the Japanese mathematician Iwasawa, located on the border of classical number theory and modern algebraic geometry. This fusion of seemingly distant mathematical disciplines is called arithmetic algebraic geometry. Andrew challenged Fermat's problem, relying precisely on this synthetic theory, difficult even for many professional mathematicians.
After completing graduate school, Wiles accepted a position at Princeton University, where he still works. He is married and has three daughters, two of whom were born "during the seven-year process of the first version of the proof." 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.
The events themselves around the proof unfolded quite dramatically. This exciting scenario could be called “fermatist – professional mathematician”.
Indeed, Andrew dreamed of proving Fermat's theorem since his youth. But, unlike the overwhelming majority of Fermatists, it was clear to him that for this it was necessary to master entire layers of the most complex mathematics. Moving towards his goal, Andrew graduates from the Faculty of Mathematics at the famous Cambridge University and begins to specialize in modern number theory, which is at the intersection with algebraic geometry.
The idea of storming the shining peak is quite simple and fundamental - the best possible ammunition and careful development of the route.
As a powerful tool for achieving the goal, the Iwasawa theory, developed by Wiles himself and already familiar to him, which has deep historical roots, is chosen. This theory generalized Kummer's theory, historically the first serious mathematical theory to attack Fermat's problem, which appeared back in the 19th century. In turn, the roots of Kummer’s theory lie in the famous theory of the legendary and brilliant romantic revolutionary Evariste Galois, who died at the age of twenty-one in a duel in defense of a girl’s honor (pay attention, remembering the story with Taniyama, to the fatal role of beautiful ladies in the history of mathematics) .
Wiles is completely immersed in proof, even stopping participation in scientific conferences. And as a result of a seven-year retreat from the mathematical community at Princeton, in May 1993, Andrew put an end to his text - the job was done.
It was at this time that an excellent opportunity presented itself to notify the scientific world about his discovery - already in June a conference was to be held in his native Cambridge on precisely the desired topic. Three lectures at the Cambridge Institute by Isaac Newton excite not only the mathematical world, but also the general public. At the end of the third lecture, June 23, 1993, Wiles announces the proof of Fermat's Last Theorem. The proof is full of a whole bunch of new ideas, such as a new approach to the Taniyama-Shimura-Weil conjecture, a far advanced theory of Iwasawa, a new “deformation control theory” of Galois representations. The mathematical community is eagerly waiting for the text of the proof to be reviewed by experts in arithmetic algebraic geometry.
This is where the dramatic turn comes. Wiles himself, in the process of communicating with reviewers, discovers a gap in his evidence. The crack was caused by the “deformation control” mechanism he himself invented - the supporting structure of the proof.
The gap is revealed a couple of months later by Wiles' line-by-line explanation of his proof to Princeton faculty colleague Nick Katz. Nick Katz, having been on friendly terms with Andrew for a long time, recommends that he collaborate with the young promising English mathematician Richard Taylor.
Another year of hard work passes, associated with the study of an additional weapon for attacking an intractable problem - the so-called Euler systems, independently discovered in the 80s by our compatriot Viktor Kolyvagin (already a long time working at the University of New York) and Thain.
And here's a new test. Not completed, but still very impressive, the result of Wiles’ work was reported by him to the International Congress of Mathematicians in Zurich at the end of August 1994. Wiles fights hard. Literally before the report, according to eyewitnesses, he was feverishly writing something else, trying to maximally improve the situation with the “sagging” evidence.
After this intriguing audience of the world's leading mathematicians, Wiles's report, the mathematical community “exhales joyfully” and sympathetically applauds: it’s okay, guy, no matter what happens, but he has advanced science, showing that in solving such an impregnable hypothesis one can successfully advance, which no one has ever done before I didn't even think about doing it. Another Fermatist, Andrew Wiles, could not take away the secret dream of many mathematicians about proving Fermat's theorem.
It is natural to imagine Wiles's condition at that time. Even the support and friendly attitude of his colleagues could not compensate for his state of psychological devastation.
And so, just a month later, when, as Wiles writes in the introduction to his final Annals article with the final proof, “I decided to take one last look at Eulerian systems in an attempt to revive this argument for proof,” it happened. Wiles had a flash of insight on September 19, 1994. It was on this day that the gap in the proof was closed.
Then things moved at a rapid pace. Already established collaboration with Richard Taylor in the study of the Eulerian systems of Kolyvagin and Thain allowed the proof to be finalized in the form of two large papers in October.
Their publication, which filled the entire issue of the Annals of Mathematics, followed in November 1994. All this caused a new powerful information surge. The story of Wiles' proof received enthusiastic press in the United States, a film was made and books were published about the author of a fantastic breakthrough in mathematics. In one assessment of his own work, Wiles noted that he had invented the mathematics of the future.
(I wonder if this is so? Let us just note that with all this information storm there was a sharp contrast with the almost zero information resonance in Russia, which continues to this day).
Let’s ask ourselves a question: what is the “internal kitchen” of obtaining outstanding results? After all, it is interesting to know how a scientist organizes his work, what he focuses on in it, and how he determines the priorities of his activities. What can be said about Andrew Wiles in this sense? And unexpectedly it turns out that in the modern era of active scientific communication and a collective style of work, Wiles had his own view on the style of working on super problems.
Wiles achieved his fantastic result on the basis of intensive, continuous, many years of individual work. The organization of its activities, speaking in official language, was of an extremely unplanned nature. This categorically could not be called an activity within the framework of a specific grant, for which it is necessary to regularly report and, again, each time plan to obtain certain results by a certain date.
Such activity outside society, which did not involve direct scientific communication with colleagues even at conferences, seemed to contradict all the canons of the work of a modern scientist.
But it was individual work that made it possible to go beyond the already established standard concepts and methods. This style of work, closed in form and at the same time free in essence, made it possible to invent new powerful methods and obtain results of a new level.
The problem facing Wiles (the Taniyama-Shimura-Weil conjecture) was not even among the closest peaks that could be conquered by modern mathematics in those years. At the same time, none of the specialists denied its enormous significance, and nominally it was in the “mainstream” of modern mathematics.
Thus, Wiles’ activities were of a distinctly non-systemic nature and the result was achieved thanks to strong motivation, talent, creative freedom, will, more than favorable material conditions for working at Princeton and, most importantly, mutual understanding in the family.
Wiles' proof, which appeared like a bolt from the blue, became a kind of test for the international mathematical community. The reaction of even the most progressive part of this community as a whole turned out to be, oddly enough, quite neutral. After the emotions and delight of the first time after the appearance of the landmark evidence subsided, everyone calmly continued their business. Specialists in arithmetic algebraic geometry slowly studied the “mighty proof” in their narrow circle, while the rest plowed their mathematical paths, diverging, as before, further and further from each other.
Let's try to understand this situation, which has both objective and subjective reasons. Objective factors of non-perception, oddly enough, have roots in the organizational structure of modern scientific activity. This activity is like a skating rink going down a sloping road and possessing colossal inertia: its own school, its own established priorities, its own sources of funding, etc. All this is good from the point of view of an established reporting system to the grant giver, but it makes it difficult to raise your head and look around: what is actually important and relevant for science and society, and not for the next portion of a grant?
Then - again - you don’t want to get out of your cozy hole, where everything is so familiar, and climb into another, completely unfamiliar hole. It is not known what to expect there. Moreover, it is obviously clear that they don’t give money for intrusion.
It is quite natural that none of the bureaucratic structures organizing science in different countries, including Russia, have drawn conclusions not only from the phenomenon of Andrew Wiles’ proof, but also from the similar phenomenon of Grigory Perelman’s sensational proof of another, also famous mathematical problem.
The subjective factors of the neutrality of the reaction of the mathematical world to the “event of the millennium” lie in quite prosaic reasons. The proof is indeed extraordinarily complex and lengthy. To a non-specialist in arithmetic algebraic geometry, it appears to consist of a layering of terminology and constructions of the most abstract mathematical disciplines. It seems that the author did not at all set a goal for him to be understood by as many interested mathematicians as possible.
This methodological complexity, unfortunately, is present as an inevitable cost of the great proofs of recent times (for example, the analysis of Grigory Perelman’s recent proof of the Poincaré conjecture continues to this day).
The complexity of perception is further enhanced by the fact that arithmetic algebraic geometry is a very exotic subfield of mathematics, causing difficulties even for professional mathematicians. The matter was also aggravated by the extraordinary synthetic nature of Wiles's proof, which used a variety of modern tools created by a large number of mathematicians in recent years.
But we must take into account that Wiles was not faced with the methodological task of explanation - he was constructing a new method. What worked in the method was precisely the synthesis of Wiles’s own brilliant ideas and a conglomerate of the latest results from various mathematical directions. And it was precisely such a powerful structure that rammed the impregnable problem. The proof was not an accident. The fact of its crystallization was fully consistent with both the logic of the development of science and the logic of knowledge. The task of explaining such a super-proof seems to be an absolutely independent, very difficult, although very promising problem.
You can test public opinion yourself. Try asking questions to mathematicians you know about Wiles' proof: who understood? Who understood at least the basic ideas? Who wanted to understand? Who felt that this was new mathematics? The answers to these questions seem rhetorical. And you are unlikely to meet many people who want to break through the palisade of special terms and master new concepts and methods in order to solve just one very exotic equation. And why is it necessary to study all this for the sake of this particular task?!
Let me give you a funny example. A couple of years ago, the famous French mathematician, Fields laureate, Pierre Deligne, a leading specialist in algebraic geometry and number theory, when asked by the author about the meaning of one of the key objects of Wiles’s proof - the so-called “ring of deformations” - after half an hour of reflection, said that it was not completely understands the meaning of this object. Ten years have already passed since the proof by this point.
Now we can reproduce the reaction of Russian mathematicians. The main reaction is its almost complete absence. This is mainly due to Wiles' "heavy" and "unusual" mathematics.
For example, in classical number theory you will not find such long proofs as Wiles's. As number theorists say, “a proof should be a page long” (Wiles’s proof in collaboration with Taylor in the journal version takes 120 pages).
You also cannot exclude the factor of fear for the unprofessionalism of your assessment: by reacting, you take responsibility for assessing the evidence. How to do this when you don’t know this mathematics?
The position taken by direct specialists in number theory is characteristic: “... and awe, and burning interest, and caution in the face of one of the greatest mysteries in the history of mathematics” (from the preface to the book by Paulo Ribenboim “Fermat’s Last Theorem for Amateurs” - the only one available today day to the source directly from Wiles' proof for the general reader.
The reaction of one of the most famous modern Russian mathematicians, Academician V.I. Arnold is “actively skeptical” about the proof: 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. himself in recent years. Arnold's works 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.
At the Faculty of Mechanics and Mathematics of Moscow State University, however, proof enthusiasts appear. A remarkable mathematician and popular scientist Yu.P. Soloviev (untimely departed from us) initiates the translation of E. Knapp’s book on elliptic curves with the necessary material on the Taniyama-Shimura-Weil conjecture. Alexey Panchishkin, now working in France, gave lectures at the Faculty of Mechanics and Mathematics in 2001, which served as the basis for his corresponding part with Yu.I. Manin of the excellent book on modern number theory mentioned above (published in Russian translation by Sergei Gorchinsky with editing by Alexei Parshin in 2007).
It is somewhat surprising that at the Moscow Steklov Mathematical Institute - the center of the Russian mathematical world - Wiles' proof was not discussed in seminars, but was studied only by individual specialized experts. Moreover, the proof of the already complete Taniyama-Shimura-Weil conjecture was not understood (Wiles proved only its part, sufficient to prove Fermat’s theorem). This proof was given in 2000 by a whole team of foreign mathematicians, including Richard Taylor, Wiles’ co-author on the final stage of the proof of Fermat’s theorem.
There were also no public statements, much less discussions, on the part of famous Russian mathematicians regarding Wiles’ proof. There is a rather sharp discussion between the Russian V. Arnold (“a skeptic of the method of proof”) and the American S. Lang (“an enthusiast of the method of proof”), however, traces of it are lost in Western publications. In the Russian central mathematical press, during the time that has passed since the publication of Wiles' proof, there have been no publications on the topic of the proof. Perhaps the only publication on this topic was a translation of an article by Canadian mathematician Henry Darmon, even an incomplete version of the proof, in Advances in Mathematical Sciences in 1995 (it’s funny that the complete proof has already been published).
Against this "sleepy" mathematical background, despite the highly abstract nature of Wiles's proof, some intrepid theoretical physicists included it in their area of potential interest and began to study it, hoping to sooner or later find applications of Wiles' mathematics. This cannot but rejoice, if only because this mathematics has been practically in self-isolation all these years.
Nevertheless, the problem of proof adaptation, which extremely aggravates its applied potential, remained and remains very relevant. Today, the original highly specialized text of Wiles's article and the joint paper of Wiles and Taylor has already been adapted, although only for a fairly narrow circle of professional mathematicians. This was done in the mentioned book by Yu. Manin and A. Panchishkin. They managed to successfully smooth out a certain artificiality of the original proof. In addition, the American mathematician Serge Lang, an ardent promoter of Wiles' proof (who sadly passed away in September 2005), included some of the most important constructions of the proof in the third edition of his now classic university textbook Algebra.
As an example of the artificiality of the original proof, we note that one of the particularly striking features that creates this impression is the special role of individual prime numbers such as 2, 3, 5, 11, 17, as well as individual natural numbers such as 15, 30 and 60. Among other things, it is quite obvious that the proof is not geometric in the most ordinary sense. It does not contain natural geometric images to which one could attach for a better understanding of the text. Super-powerful “terminologized” abstract algebra and “advanced” number theory purely psychologically undermine the ability to perceive proof even for a qualified mathematical reader.
One can only wonder why, in such a situation, proof experts, including Wiles himself, do not “polish” it, do not promote and popularize an obvious “mathematical hit” even in their native mathematical community.
So, in short, today the fact of Wiles’ proof is simply the fact of the proof of Fermat’s theorem with the status of the first correct proof and “some kind of super-powerful mathematics” used in it.
The famous Russian mathematician of the middle of the last century, former dean of the Faculty of Mechanics and Mathematics, V.V., spoke very clearly about the powerful, but not yet applied, mathematics. Golubev:
“... according to the witty remark of F. Klein, many departments of mathematics are similar to those exhibitions of the latest models of weapons that exist at companies that manufacture weapons; with all the wit put in by the inventors, it often happens that when a real war begins, these new products turn out to be unusable for one reason or another... The modern teaching of mathematics presents exactly the same picture; students are given very advanced and powerful means of mathematical research into their hands..., but then students cannot bear any idea of where and how these powerful and ingenious methods can be applied in solving the main task of all science: in understanding the world around us and in influencing it is the creative will of man. At one time A.P. Chekhov said that if in the first act of a play there is a gun hanging on the stage, then it is necessary that at least in the third act it be fired. This remark is fully applicable to the teaching of mathematics: if any theory is presented to students, then it is necessary to show sooner or later what applications can be made from this theory, primarily in the field of mechanics, physics or technology and in other areas.”
Continuing this analogy, we can say that Wiles’ proof represents extremely favorable material for studying a huge layer of modern fundamental mathematics. Here students can be shown how the problem of classical number theory is closely related to such branches of pure mathematics as modern algebraic number theory, modern Galois theory, p-adic mathematics, arithmetic algebraic geometry, commutative and non-commutative algebra.
It would be fair if Wiles’s confidence that the mathematics he invented—mathematics of a new level—was confirmed. And I really don’t want this really very beautiful and synthetic mathematics to suffer the fate of an “unfired gun.”
And yet, let us now ask the question: is it possible to describe Wiles’ proof in sufficiently accessible terms for a wide interested audience?
From the point of view of experts, this is an absolute utopia. But let’s try anyway, guided by the simple consideration that Fermat’s theorem is a statement only about integer points of our ordinary three-dimensional Euclidean space.
We will sequentially substitute points with integer coordinates into Fermat’s equation.
Wiles finds the optimal mechanism for recalculating integer points and testing them to satisfy the equation of Fermat’s theorem (after introducing the necessary definitions, such a recalculation will precisely correspond to the so-called “modularity property of elliptic curves over the field of rational numbers”, described by the Taniyama-Shimura-Weil conjecture).
The recalculation mechanism is optimized with the help of a remarkable discovery by the German mathematician Gerhard Frey, who connected a potential solution of the Fermat equation with an arbitrary exponent with another, completely different equation. This new equation is given by a special curve (called Frey's elliptic curve). This Frey curve is given by a very simple equation:
The surprise of Frey's idea was the transition from the number-theoretic nature of the problem to its “hidden” geometric aspect. Namely: Frey associated with every solution of Fermat’s equation, that is, numbers satisfying the relation
the above curve. Now it remains to show that such curves do not exist for . In this case, Fermat's last theorem would follow. This is exactly the strategy that Wiles chose in 1986, when he began his enchanting assault.
Frey’s invention at the time of Wiles’s “start” was quite fresh (the year 1985) and also echoed the relatively recent approach of the French mathematician Helleguarche (the 1970s), who proposed using elliptic curves to find solutions to Diophantine equations, i.e. equations similar to Fermat's equation.
Let's now try to look at the Frey curve from a different point of view, namely, as a tool for recalculating integer points in Euclidean space. In other words, our Frey curve will play the role of a formula that determines the algorithm for such a recalculation.
In this context, we can say that Wiles invents tools (special algebraic constructions) to control this recalculation. As a matter of fact, this subtle toolkit of Wiles constitutes the central core and main complexity of the proof. It is in the manufacture of these instruments that Wiles's main sophisticated algebraic discoveries, which are so difficult to comprehend, arise.
But still, the most unexpected effect of the proof, perhaps, is the sufficiency of using only one “Freevian” curve, represented by a completely simple, almost “school” dependence. Surprisingly, using only one such curve is sufficient to test all points in three-dimensional Euclidean space with integer coordinates to see if they satisfy Fermat's Last Theorem with an arbitrary exponent.
In other words, using just one curve (though it has a specific form), understandable to an ordinary high school student, turns out to be equivalent to constructing an algorithm (program) for sequential recalculation of whole points of ordinary three-dimensional space. And not just a recalculation, but a recalculation with simultaneous testing of the whole point for “its satisfaction” with Fermat’s equation.
It is here that the phantom of Pierre de Fermat himself arises, since with such a recalculation what is usually called Fermat’s “Ferma’t descent,” or reduction (or “method of infinite descent”) comes to life.
In this context, it immediately becomes clear why Fermat himself could not prove his theorem for objective reasons, although he could well “see” the geometric idea of its proof.
The fact is that the recalculation takes place under the control of mathematical tools that have no analogues not only in the distant past, but also unknown before Wiles even in modern mathematics.
The most important thing here is that these tools are “minimal”, i.e. they cannot be simplified. Although this “minimalism” in itself is very difficult. And it was Wiles’s awareness of this non-trivial “minimality” that became the decisive final step of the proof. This was exactly the “outbreak” on September 19, 1994.
Some problem that causes dissatisfaction still remains here - Wiles does not explicitly describe this minimal construction. Therefore, those interested in Fermat's problem still have interesting work to do - a clear interpretation of this “minimality” is necessary.
It is possible that this is where the geometry of the “algebraized” proof should be hidden. It is possible that it was precisely this geometry that Fermat himself felt when he made the famous entry in the narrow margins of his treatise: “I have found a truly remarkable proof …”.
Now let’s move directly to the virtual experiment and try to “dig” into the thoughts of mathematician-lawyer Pierre de Fermat.
The geometric image of Fermat’s so-called little theorem can be represented as a circle rolling “without slipping” along a straight line and “winding” whole points around itself. The equation of Fermat's little theorem in this interpretation also receives a physical meaning - the meaning of the law of conservation of such motion in one-dimensional discrete time.
You can try to transfer these geometric and physical images to the situation when the dimension of the problem (the number of variables in the equation) increases and the equation of Fermat’s little theorem transforms into the equation of Fermat’s big theorem. Namely: let us assume that the geometry of Fermat’s last theorem is represented by a sphere rolling along a plane and “winding” entire points on this plane around itself. It is important that this rolling should not be arbitrary, but “periodic” (mathematicians also say “cyclotomic”). The periodicity of rolling means that the linear and angular velocity vectors of a sphere rolling in the most general manner after a certain fixed time (period) are repeated in magnitude and direction. This periodicity is similar to the periodicity of the linear speed of rolling a circle along a straight line, modeling the “small” Fermat equation.
Accordingly, the “large” Fermat equation takes on the meaning of the law of conservation of the above-mentioned motion of the sphere already in two-dimensional discrete time. Let us now take the diagonal of this two-dimensional time (it is in this step that all the difficulty lies!). This extremely tricky and turns out to be the only diagonal is the equation of Fermat’s Last Theorem, when the exponent of the equation is exactly two.
It is important to note that in a one-dimensional situation - the situation of Fermat's little theorem - there is no need to find such a diagonal, since time is one-dimensional and there is no reason to take a diagonal. Therefore, the degree of a variable in the equation of Fermat’s little theorem can be arbitrary.
So, quite unexpectedly, we get a bridge to the “physicalization” of Fermat’s great theorem, that is, to the appearance of its physical meaning. How can one not remember that Fermat was no stranger to physics.
By the way, the experience of physics also shows that the laws of conservation of mechanical systems of the above type are quadratic in the physical variables of the problem. And finally, all this is quite consistent with the quadratic structure of the laws of conservation of energy of Newtonian mechanics, known from school.
From the point of view of the above “physical” interpretation of Fermat’s last theorem, the property of “minimality” corresponds to the minimality of the degree of the conservation law (this is two). And the reduction of Fermat and Wiles corresponds to the reduction of the laws of conservation of recalculation of points to the law of the simplest form. This simplest (minimal in complexity) recalculation, both geometrically and algebraically, is represented by the rolling of a sphere on a plane, since a sphere and a plane are “minimal,” as we completely understand, two-dimensional geometric objects.
The whole complexity, which at first glance is missing, lies in the fact that an accurate description of such a seemingly “simple” movement of the sphere is not at all easy. The fact is that the “periodic” rolling of the sphere “absorbs” a bunch of so-called “hidden” symmetries of our three-dimensional space. These hidden symmetries are caused by non-trivial combinations (compositions) of the linear and angular motion of the sphere - see Fig. 1.
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It is for the exact description of these hidden symmetries, geometrically encoded by such a tricky rolling of the sphere (points with integer coordinates “sit” at the nodes of the drawn lattice), that Wiles’ algebraic constructions are required.
In the geometric interpretation shown in Fig. 1, the linear movement of the center of the sphere “counts” whole points on the plane, and its angular (or rotational) movement provides the spatial (or vertical) component of the recalculation. The rotational motion of the sphere cannot be immediately “seen” in the arbitrary rolling of the sphere along the plane. It is the rotational motion that corresponds to the hidden symmetries of Euclidean space mentioned above.
The Frey curve introduced above precisely “encodes” the most aesthetically beautiful recalculation of whole points in space, reminiscent of movement along a spiral staircase. Indeed, if you follow the curve that a certain point on the sphere sweeps over one period, you will find that our marked point sweeps the curve shown in Fig. 2, resembling a “double spatial sinusoid” - a spatial analogue of the graph. This beautiful curve can be interpreted as a plot of the "minimum" of the (i.e.) Frey curve. This is the schedule of our testing recalculation.
Having connected some associative perception of this picture, to our surprise we will find that the surface limited by our curve is strikingly similar to the surface of the DNA molecule - the “corner brick” of biology! It is perhaps no coincidence that the terminology for DNA-encoding constructs from Wiles' proof is used in Singh's book Fermat's Last Theorem.
Let us emphasize once again that the decisive point in our interpretation is the fact that the analogue of the conservation law for Fermat’s little theorem (its degree can be arbitrarily large) turns out to be the equation of Fermat’s Great Theorem precisely in the case . It is this effect of “minimality of the degree of conservation law for the rolling of a sphere on a plane” that corresponds to the statement of Fermat’s Last Theorem.
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It is quite possible that Fermat himself saw or felt these geometric and physical images, but could not imagine that they were so difficult to describe from a mathematical point of view. Moreover, he could not imagine that to describe such, although non-trivial, but still quite transparent geometry, another three hundred and fifty years of work of the mathematical community would be required.
Now let's build a bridge to modern physics. The geometric image of Wiles's proof proposed here is very close to the geometry of modern physics, which is trying to get to the mystery of the nature of gravity - the quantum general theory of relativity. To confirm this, at first glance unexpected, interaction between Fermat’s Last Theorem and Big Physics, let’s imagine that the rolling sphere is massive and “pushes” the plane beneath it. The interpretation of this “pushing” in Fig. 3 is strikingly reminiscent of the well-known geometric interpretation of Einstein’s general theory of relativity, which describes precisely the “geometry of gravity.”
And if we also take into account the present discretization of our picture, embodied by a discrete integer lattice on a plane, then we actually observe “quantum gravity” with our own eyes!
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It is on this major “unifying” physico-mathematical note that we will end our “cavalry” attempt to give a visual interpretation of Wiles’ “super-abstract” proof.
Now, perhaps, it should be emphasized that in any case, whatever the correct proof of Fermat’s theorem, it must in one way or another use the constructions and logic of Wiles’ proof. It is simply impossible to bypass all this due to the mentioned “minimality property” of Wiles’ mathematical tools used for the proof. In our “geometric-dynamical” interpretation of this proof, this “minimality property” provides the “minimum necessary conditions” for a correct (i.e., “convergent”) construction of a testing algorithm.
On the one hand, this is a huge disappointment for amateur farmers (if, of course, they find out about it; as they say, “the less you know, the better you sleep”). On the other hand, the natural “unsimplification” of Wiles’s proof formally makes life easier for professional mathematicians - they may not read periodically emerging “elementary” proofs from amateur mathematics, citing the lack of correspondence with Wiles’s proof.
The general conclusion is that both need to “strain” and understand this “savage” proof, essentially comprehending “all mathematics.”
What else is important not to miss when summing up this entire unique story that we have witnessed? The strength of Wiles's proof is that it is not simply a formal logical argument, but represents a broad and powerful method. This creation is not a separate tool for proving one single result, but an excellent set of well-chosen tools that allows you to “split” a wide variety of problems. It is also fundamentally important that when we look down from the height of the skyscraper at Wiles’s proof, we will see all the previous mathematics. The pathos is that it will not be a “patchwork”, but a panoramic vision. All this speaks not only of the scientific, but also of the methodological continuity of this truly magical evidence. All that remains is “just nothing” - just understand it and learn to apply it.
I wonder what our contemporary hero Wiles is doing today? There is no special news about Andrew. He, naturally, received various awards and prizes, including the famous German Wolfskehl Prize, which was depreciated during the first civil war. In all the time that has passed since the triumph of the proof of Fermat’s problem until today, I managed to notice only one, albeit as always large, article in the same “Annals” (co-authored with Skinner). Maybe Andrew is hiding again in anticipation of a new mathematical breakthrough, for example, the so-called “abc” conjecture - recently formulated (by Masser and Oesterle in 1986) and considered the most important problem in number theory today (it is the “problem of the century” in the words of Serge Lang ).
Much more information about Wiles' co-author on the final part of the proof, Richard Taylor. He was one of the four authors of the proof of the full Taniyama-Shmura-Weil conjecture and was a strong contender for the Fields Medal at the 2002 Chinese Mathematical Congress. However, he did not receive it (then only two mathematicians received it - the Russian mathematician from Princeton Vladimir Voevodsky “for the theory of motives” and the Frenchman Laurent Laforgue “for an important part of the Langlands program”). Taylor published a considerable number of remarkable works during this time. And recently, Richard achieved a new great success - he proved a very famous conjecture - the Tate-Saito conjecture, also related to arithmetic algebraic geometry and generalizing the results of German. 19th century mathematician G. Frobenius and 20th century Russian mathematician N. Chebotarev.
Let's finally dream up a little. Perhaps the time will come when mathematics courses in universities, and even in schools, will be adjusted to Wiles' methods of proof. This means that Fermat's Last Theorem will become not only a model mathematical problem, but also a methodological model for teaching mathematics. Using her example, it will be possible to study, in fact, all the main branches of mathematics. Moreover, future physics, and maybe even biology and economics, will begin to rely on this mathematical apparatus. But what if?
It seems that the first steps in this direction have already been taken. This is evidenced, for example, by the fact that the American mathematician Serge Lang included the main constructions of Wiles' proof in the third edition of his classic manual on algebra. The Russians Yuri Manin and Alexey Panchishkin go even further in the aforementioned new edition of their “Modern Theory of Numbers,” setting out in detail the proof itself in the context of modern mathematics.
And how can one not exclaim now: Fermat’s great theorem is “dead” - long live Wiles’ method!