Chapter 17 The Charm of Numbers: Reading Notes on Fermat's Last Theorem

The process of seeking the proof of Fermat's last theorem has involved the most intelligent people on the planet, full of desperate resistance, unexpected turnaround, forbearing patience, and brilliant spirituality. unsolved case The process of Fermat's last theorem itself from proposing to proving is an out-and-out thriller novel. A reader who leaves notes in the margins of books he has read.Who else would care but himself? However, after the death of the Frenchman Fermat, the notes he wrote on a book "Arithmetic" did not disappear with it.Realizing that the scribbled handwriting might have value, the eldest son spent five years sorting it out, and then printed a special edition of Arithmetic with his father's marginal notes, which contained a series of theorems.

In the margin near Question 8, Fermat writes these words: "It is impossible to write a cubic number as the sum of two cubes; or a 4th power as the sum of two 4th powers; or, in general, a power higher than 2 as the two sum of the same power." This mischievous genius wrote an additional comment later: "I have a very beautiful proof of this proposition, which is too small to fit here." Fermat wrote these lines around 1637, and these clues discovered by fluke became the misfortune of all subsequent mathematicians.A theorem that can be understood by a high school student has become the biggest unsolved case in the mathematics world, and has tortured the smartest minds in the world for 358 years.Generations of mathematical geniuses have challenged this conjecture.

Euler, one of the greatest mathematicians of the 18th century, found Fermat crypticly describing a proof to the 4th power elsewhere in that particular edition of Arithmetic.Euler perfected this ambiguous proof in detail, and proved that the power of 3 has no solution.But after his breakthrough, there are still countless powers to prove. When Sophie Germain, Legendre, Dirichlet, Gabriel Lame and other Frenchmen made another breakthrough, nearly 200 years had passed since Fermat wrote that theorem, and they had just The 5th and 7th powers are proved. In fact, Lame had announced that he was about to prove Fermat's Last Theorem, and another mathematician, Cauchy, followed suit and said that he would publish a complete proof.However, a letter shattered their confidence: the German mathematician Kummer saw that the two Frenchmen were heading towards the same logical dead end.

While putting the two mathematicians to shame, Kummer also proved that a complete proof of Fermat's Last Theorem was impossible with the mathematical methods of the time.This is a glorious page of mathematical logic and a huge blow to a whole generation of mathematicians. In the 20th century, mathematics began to turn into various fields of study and made extraordinary progress. In 1908, the German industrialist Wolfskell set up a bonus for those who might conquer Fermat's last theorem in the future, but an unknown mathematician seemed to destroy everyone's hopes: Kurt Gödel proposed the impossibility A deterministic theorem, a brutal formulation of Fermat's Last Theorem - a proposition without any proof.

Despite Gödel's deadly warnings, and despite three centuries of heroic failure, some mathematicians continued to devote themselves to the problem, risking their lives for nothing.With the advent of computers after World War II, massive calculations were no longer a problem.With the help of computers, mathematicians proved Fermat's Last Theorem for values ​​up to 500, then up to 1,000, then up to 10,000, and by the 1980s, this range had increased to 25,000, then up to 4 million. However, this kind of success is only superficial. Even if the scope is further improved, it can never be proved to infinity, and it cannot be claimed to have proved the entire theorem.Solving the case seems far away.

The last hero has appeared. In 1963, Andrew Wiles, who was only ten years old, encountered Fermat's Last Theorem in a book called "The Big Problem", and knew that he would never give up on it and had to solve it. In the 1970s, he was studying elliptic equations at Cambridge University, which seemed to have nothing to do with Fermat's last theorem. At this time, two Japanese mathematicians had proposed the Taniyama-Shimura conjecture, which unified the elliptic equation and the modular form that Wiles was studying.It seems that it has nothing to do with Fermat's last theorem.

In the 1980s, several mathematicians combined the most important problems of the 17th century with the most meaningful problems of the 20th century, and found the key to prove Fermat's last theorem: as long as the Taniyama-Shimura conjecture can be proved, it will automatically prove Fermat's last theorem. Horse's last theorem. The dawn is ahead, but no one has confidence in the arrival of the dawn. The Taniyama-Shimura conjecture has been studied for 30 years and all ended in failure. Now it is connected with Fermat's last theorem, and there is no last hope Yes, because anything that would lead to a solution to Fermat's Last Theorem is by definition impossible - that's pretty much a given.

Even Ken Ribet, the key figure in the discovery of the key, was pessimistic: "I haven't really bothered trying to prove it, or even thought of trying." Most other mathematicians, including Andrew Wiles, The mentor, John Coates, believed that doing this proof would be futile: "I must admit that I think it is probably impossible to see it proved in my lifetime." Except Andrew Wiles. The great logician David Hilbert was once asked why he didn't try to prove Fermat's Last Theorem, and he replied, "I don't have that much time to waste on a thing that might fail."

But Andrew Wiles will.He realized that his chances were slim, but even if he failed to prove Fermat's Last Theorem, he felt that his efforts would not be in vain.It took him 18 months to gather the necessary weapons for the battle ahead, and he came up with a full estimate: Any serious attempt at this proof would likely take 10 years of dedicated effort. Wiles gave up all the work not directly related to the proof of Fermat's last theorem, and in a state of complete secrecy, he launched a lonely challenge to this mystery that has plagued the world's wise men for more than three hundred years. His wife is the only one who knows that he is working on it. The person who studied the Fermat problem.

After seven years of hard work, Andrew Wiles completed the proof of the Taniyama-Shimura conjecture. On June 23, 1993, at the Newton Institute in Cambridge, he started the most important mathematics lecture of this century, and everyone who contributed to the proof of Fermat's Last Theorem was actually in the room at the scene, two hundred people Mathematicians were stunned to see, for the first time in more than 300 years, Fermat's challenge conquered. Wiles wrote the conclusion of Fermat's Last Theorem, then turned to the audience and said calmly, "I think I'll end here." Newspaper headlines. People magazine named him one of its "25 Most Glamorous People of the Year" along with Princess Diana and Oprah, and a fashion house tapped the suave genius to advertise their new line of menswear.

But the matter did not end here, and the subsequent development was still like a thriller novel. The unsolved case was solved, but the criminal was not easily caught.Wiles' 200-page manuscript was submitted to the "Mathematical Invention" magazine, and a complex review process began.It is a very large argument, intricately constructed from hundreds of mathematical calculations through thousands of logical links.As long as there is a calculation error or a chain link is not connected properly, the entire proof may lose its value. A problem worth solving proves its worth by fighting back.During the demanding review process, the reviewers encountered what appeared to be a minor issue.The essence of the problem is that Wiles cannot be made to guarantee that a certain method will work, as he originally imagined.He has to strengthen his proof. As time passed, the problem remained unresolved, and the whole world began to doubt Wiles. Fourteen months on, he was ready to admit defeat publicly and issue a statement that proved to be flawed.In his last moments, on a Monday morning, September 19, 1995, he decided to take one last look, trying to determine exactly why that method wasn't working. A sudden inspiration brought his suffering to an end: Although that method didn't quite work, it only needed to make another theory he had given up work, and the correct answer could emerge from the ruins—two separate Inadequate solutions combine to perfectly complement each other. For a full 20 minutes, Wiles stared at the result in disbelief, and then, there was a great sense of loss that there was nothing else to do. A hundred years ago, the Wolfskell Prize dedicated to Fermat's Last Theorem set a deadline of September 13, 2007.Like all thrillers, the bomb was defused at the last moment. legend "Fermat's Last Theorem" is not only a thrilling novel, but also a martial arts novel, which inspires legendary stories passed down through the ages by top masters. The rivers and lakes in the mathematics world belong to young people.Teenage heroes show their talents here. When Kurt Gödel proposed his undecidability theorem, he was only 25 years old, and he pushed his contemporaries into the abyss of despair; Norway's Abel was 19 He made his greatest contribution to mathematics at the age of 8, and died in poverty 8 years later. The French mathematician Emmett commented that "the ideas he left can be used by mathematicians for 500 years"; , Andrew Wiles did not complete Fermat's last theorem until he was 40 years old. Others think that he should be the age when his ideas are exhausted. "Young people should prove theorems, but old people should write books." British mathematician Hardy said, "Compared with other arts or sciences, mathematics is a young man's game." Which territory is more suitable for young people? Write a legend?Mathematicians have the lowest average age at election to membership in the Royal Society. The stories surrounding Fermat's last theorem are beyond the imagination of the best screenwriters. In January 1954, Goro Shimura, a young mathematician at the University of Tokyo, went to the department library to borrow a book. To his surprise, the book was borrowed by a man named Toyo Taniyama.Shimura wrote a letter to this unfamiliar alumnus, and a few days later, he received a postcard from the other party, and Taniyama told him that he was doing the same calculation and was stuck in the same place. A tacit understanding of surprise emerged immediately, and the two began a sympathetic cooperation. "He was born with a special ability to make many mistakes, especially mistakes in the right direction." Shimura commented on his partner. On November 17, 1958, Gu Shan, who was just engaged, this absent-minded genius chose to commit suicide.A few weeks later, his fiancée also took her own life, writing: "Now that he is gone, I must be with him too." In his suicide note, Taniyama apologized to his colleagues for all the troubles caused by his suicide, and the many fundamental ideas on mathematics he left behind became the only key to unlock Fermat's last theorem: Taniyama- Shimura guessed. Thirty years later, his partner Shimura, who witnessed their conjecture confirmed, told reporters with restraint and self-respecting calm: "I told you it was right." He still has the postcard that Taniyama sent him for the first time. German industrialist Wolfskell was not a gifted mathematician, but a most incredible event forever linked him to Fermat's last theorem. Obsessed with and rejected by a beautiful woman, Wolfskell despairs.He decides to kill himself, and sets a date to kill himself, ready to shoot himself in the head when the clock strikes midnight.Wolfskell took care of every detail: business affairs were handled, a will was written, and letters were written to all friends and family. His high efficiency allowed everything to be done just before the midnight deadline.In order to kill the last few hours, he went to the library to read mathematics books: a paper on the proof of Fermat's last theorem...He picked up the pen unconsciously, and calculated line by line... Then, it was dawn. Wolfskell was immensely proud of having found and corrected a flaw in his thesis. The original despair and sadness disappeared, and mathematics called him back from death. In 1908, Wolfskell, who had enjoyed his life, wrote his new will: a large part of his property was used as a prize, and it was stipulated that anyone who could prove Fermat's last theorem would be rewarded with a prize of 100,000 Mark, over £1 million in today's currency. It was his way of repaying the insane problem that had saved his life. The French mathematician Galois was involved in an affair.The woman he was in love with was actually engaged. The gentleman found out about his fiancée's infidelity and angrily challenged Galois to a duel. The opponent is one of the best shooters in France, and Galois is very aware of his strength: let alone shooting, he only performs mathematical calculations in his head, and disdains to write down the arguments clearly on paper. Many of his mathematical achievements were not valued and recognized by the French Academy of Sciences.The night before the duel he believed to be his last, his last chance to put his thoughts on paper. He stayed up all night and wrote all the theorems that existed in his head.In the complex algebraic formula, the woman's name is hidden from time to time, and there is a desperate exclamation-"I have no time, I have no time!" The next day, May 30, 1832, Galois died in a duel. When his scribbled manuscript was handed over to some mathematicians in contact in Europe, the genius ideas burst out in those calculations made the experts discover that one of the most outstanding mathematicians in the world was killed when he was 20 years old. Study mathematics for only 5 years. Galois gave a complete and thorough description of the solution of the quintic equation in the manuscript, and the core part of his calculus is the idea called "group theory", which he developed into a method that can overcome previously unsolvable problems. powerful tool for problems. Galois' last night's work would form the basis for Andrew Wiles' proof of the Taniyama-Shimura conjecture a century and a half later. On June 27, 1997, Andrew Wiles, who met the terms of the Wolfskell Committee and defeated the Fermat challenge, received a prize worth $50,000. Yes, Fermat's Last Theorem was officially solved.Wiles took all the breakthrough work in number theory of the 20th century and fused them into a proof-of-magnitude. People have reconsidered the line of additional commentary Fermat wrote: "I have a very beautiful proof of this proposition, and the space here is too small to write." What is certain is that a few centuries ago, Fermat did not Invented the modular form used by Andrew Wiles to prove the last theorem, the Taniyama-Shimura conjecture, the Galois group theory, and the Kolivakin-Fletcher method. So, what method did Fermat himself use to prove his conjecture?Was that just a flawed proof, or was he building on a 17th-century trick and involving another method that no mathematician had discovered for hundreds of years?We'll never have a chance to know. "That special long exploration is now over, and my mind is at peace." Andrew Wiles said. The legend seems to have ended, but in fact the greater legend was hidden forever 358 years ago. math In 212 BC, the Roman army invaded Syracuse. Archimedes, who was nearly 80 years old, was concentrating on studying a geometric figure in the sand. He neglected to answer a Roman soldier's question and was stabbed to death by a spear. Sophie Germain, a Parisian girl in the 18th century, read this chapter in a book called "The History of Mathematics", and came to the conclusion that if a person is so obsessed with a geometric problem that leads to his death , then mathematics must be the most fascinating subject in the world. She was immediately fascinated by this most fascinating subject, often working late into the night, studying the writings of Euler and Newton.Her parents confiscated her candles and clothes and removed everything that could keep her warm to prevent her from continuing her studies.She continued studying with a hidden candle and wrapped herself in a sheet, even as the ink froze in the inkwell.In the end her parents compromised. In that era full of prejudice and machismo, she pretended to be "Mr. LeBlanc" and studied at the Paris Polytechnic Institute that only accepted men through letters. In this capacity, she communicated with "King of Mathematicians" Gauss to discuss Fermat University theorem. When Napoleon invaded Prussia in 1806, Germain asked a French general to keep Gauss safe.Only then did Gauss, who received special care, know her true identity. Otherwise, her outstanding contribution to Fermat's last theorem would probably be forever remembered by that "Mr. LeBlanc". Gauss talked about the magic of mathematics in his acknowledgment letter: "Nothing has yet proved to me in such a charming and unambiguous way the attractive decision of this science that has added so much joy to my life. Not fictional." His statement was too lengthy.Let Germain's kind answer this question-when someone asked the female mathematician Hypatia in the 4th century AD why she never married, she said that she was married to truth. Like most female mathematicians who have emerged in the past two thousand years, Sophie Germain never married. Everything is number, this is the magic of mathematics. Numbers show up wonderfully in all kinds of natural phenomena.Looking at all the winding rivers in the world, Hans Henrik, a geoscientist at the University of Cambridge, found that the ratio of the actual length to the straight-line distance from the source of the river to the mouth of the river is basically close to the value of pi.Einstein proposed that this number emerged as the result of a struggle between order and disorder. In fact, as early as the 6th century BC, Pythagoras discovered the relationship between numbers and nature.He recognized that natural phenomena are governed by laws that can be described by mathematical equations.For example, he discovered in the smithy the relationship between musical harmony and number harmony: those hammers that are in harmony with each other have a simple mathematical relationship, their masses are simple ratios, or simple fractions, to each other, Like a half, a third, a quarter. Among insects, cicadas have the longest life cycle at 17 years.Does this prime number have any special meaning?According to the interpretation of biologists, the life cycle of this prime number protects it.Only two parasites can threaten it: 1-year or 17-year.And the parasites could not have survived for 17 consecutive years because there were no cicadas for them to feed on during the first 16 appearances.Therefore, the life cycle being a prime number has some evolutionary advantages.And it turns out, too: the cicada parasite has never been found. The mystery of the numbers themselves is even more exciting.A perfect number means that the sum of the factors of a number is exactly equal to the number itself, for example, the factors of 6 are 1, 2, and 3, and the sum of the latter is exactly 6, so it is a perfect number.This concept has been proposed for nearly three thousand years, but mathematicians have only discovered 30 perfect numbers, and the lovely old 6 is the smallest one.St. Augustine said: "6 is a number that is perfect in itself, not because God created everything in 6 days; the reverse is true: God created everything in 6 days because this number is perfect. " Another example is 26, which Fermat noticed was sandwiched between a square number (25 is 5 squared) and a cubic number (27 is 3 cubed).He failed to find other such numbers, so is 26 the only one?So far no one has been able to come up with proof. Saying what you say is another magic of mathematics. In the Kingdom of Mathematics, there is no such thing as a public saying that the public is right, or a woman saying that the woman is right, and there is no debate contest where pros and cons are opposed. Participants draw lots to decide their positions, and the one who wins in the end is actually a person with good eloquence. In the mathematical lexicon, mathematical proof is a powerful and rigorous concept that rises above scientific proof as understood by a physicist or chemist.Scientific proof relies on observation and understanding, and operates according to the evaluation system. If there is enough evidence to prove that a theory is "free from all reasonable doubts", then the theory is considered correct.Mathematics, on the other hand, does not depend on the evidence of fallible experiments, but on infallible logic, leading to conclusions that are indubitably correct and will never be disputed. Science provides only approximate concepts of truth, while mathematics itself is truth.Mathematics gives science a rigorous beginning, and to this infallible foundation scientists add imprecise measurements and flawed observations. So we can understand the cruelty of mathematicians. With the help of computers, some people can conclude that Fermat's last theorem is correct for powers up to 4 million, but the proposition is still not proved. There are counterexamples in this regard. 31, 331, 3331, 33331, 333331, 3333331, 33333331, after careful exploration, mathematicians have proved that these numbers are all prime numbers, so are the numbers in this form all prime numbers?Not the next number, 333333331, which can be factored into 17 times 19607843. After Fermat's last theorem, Euler also proposed a conjecture that it is impossible to write a power higher than 2 as the sum of three powers of the same power.No one has been able to prove this conjecture for more than 200 years. Afterwards, a computer search was used, but no solution was found. There is no counterexample, which is strong evidence for the establishment of this conjecture. However, prudent mathematicians will not admit Euler's conjecture because of this.Sure enough, in 1988, Naom of Harvard University discovered a solution: the 4th power of 2682440 plus the 4th power of 15365639 plus the 4th power of 18796760 is equal to the 4th power of 20615673. Relying on absolutely reliable axioms and theorems, mathematicians have built a solid mathematical building, each cornerstone is reliable, and the whole building has become the most trustworthy building in the home of human wisdom. This is the glory of mathematics. The charm of mathematics cares about the challenges to human intelligence and curiosity. Up to now, mathematics has become the loneliest science in the world.Mathematicians working on cutting-edge problems may only count in single digits if they try to find someone to talk to and search the world.But they certainly take pride in this solitude. In the face of Fermat's last theorem, mathematicians have suffered heroic failures for more than three centuries, and any mathematician involved in it runs the risk of wasting their lives for nothing.Why do they keep going on like this? If you can prove the last theorem, then you have solved a difficult problem that other colleagues have been troubled for hundreds of years, and you have succeeded where others have failed.In addition to this sense of accomplishment that is superior to others, it is the innate curiosity of human beings that cannot be restrained.The desire to solve a mathematical problem is mostly out of curiosity, and the reward is the simple but immense satisfaction of solving a difficult problem.Mathematician Titch Marsh said: "It may be of no practical use at all to find out that pi is irrational, but if we can find out, then we must not bear to try to find out." Mathematics has its applications in science and technology, but this is not what drives mathematicians.A student asked Euclid what was the use of the mathematics he was learning, and Euclid turned around and asked the servant to drive him away: "Give the boy a coin, because he wants to gain practical benefits in his studies." Hardy in " Confessions of a Mathematician" said frankly: "From a practical point of view, the value of my mathematical career is equal to zero." When Andrew Wiles knew that he would spend ten years of hard work and that the chances of cracking Fermat's last theorem were not great, he still began to calculate diligently: "Even if they don't solve the whole problem, they will be valuable mathematics. .I don't think I'm wasting my time." Math is the greatest romance. mathematician Astronomers, physicists, and mathematicians traveled across Scotland in trains.They looked out and saw a black sheep in a field.The astronomer said: "How interesting, all Scotch sheep are black." The physicist retorted: "No! Some Scotch sheep are black." The mathematician said slowly: "In Scotland there exists at least one field , at least one sheep that is black on at least one side." Ian Stewart used this joke in "Concepts of Modern Mathematics" to reveal the meticulous and strict attitude of mathematicians: a certain conclusion needs to be proved beyond doubt. So, a real mathematician never speaks out loud.Edmund Landau of the University of Göttingen was asked whether his colleague Amy Noether was really a great female mathematician, and he replied: "I can testify that she is a great mathematician, but for her It's a woman, I can't swear." Only mathematicians are qualified to say such indisputable words. In 1986, two mathematicians, Ribbett and Meschel, met by chance in a coffee shop when they were attending the International Congress of Mathematicians in Berkeley.Ribet talked about the elliptic equation he was trying to prove, and the experimental strategies he had been exploring.Mercure sipped his cappuccino while listening to Ribet's narration.He stopped his coffee abruptly and said with certainty, "Don't you understand? You've already done it! All you need to do is add some M-structured gamma-0, and that's it." Certainly, only a handful of people in the world can figure this out over a casual cup of coffee. Mathematicians are in some respects almost insincere.During his lifetime Fermat was a civil servant and also worked in the judiciary.In order to prevent people in this position from falling into corruption, the government requires judges not to participate in social activities, so he can concentrate on studying mathematical problems.But in any case, mathematics can only be regarded as his hobby, and Eric Bell called him "the king of amateur mathematicians".But some people are not satisfied with this description.When Julian Coolidge wrote the book "Mathematics of the Great Amateur Mathematician", he insisted on excluding Fermat: "He is so outstanding that he should be counted as a professional mathematician." Their tempers are just as fiery.Sophie Germain made outstanding contributions to the proof of Fermat's last theorem. She also made great achievements in the field of physics and won the gold medal of the French Academy of Sciences. Women who attended lectures at the Academy of Sciences.Under the persuasion of Gauss, the University of Göttingen was going to award her an honorary doctorate. Unfortunately, Germain had died of breast cancer at this time. When the officials issued the death certificate for Germain, they listed her as an "unemployed unmarried woman" instead of a female mathematician.And she, who made great contributions to the theory of material elasticity, did not appear in the names of the 72 experts inscribed on the Eiffel Tower.Mauzens lashed out at the event: "Those who are responsible for this ingratitude to a man so deserving of science and who, by virtue of her achievements, has acquired an enviable place in the halls of fame How ashamed people should be." A writer will never become a mathematician, but a mathematician may write very touching and temperamental words. Because they say one thing, because they are either one or the other, and because they are undisputed, mathematicians have the openness and openness that are different from ordinary people who are willing to bet and admit defeat. In "A Beautiful Mind," a group of mathematicians present pens to John Nash in the hall as a way of paying tribute.This scene reflects the unique quack morality and ethics in the kingdom of mathematics. In order to encourage the proof of Fermat's Last Theorem, the French Academy of Sciences has established a series of awards and huge prize money. In 1847, Gabriel Lame took the podium of the Academy of Sciences and confidently predicted that in a few weeks he would publish a complete proof of Fermat's Last Theorem in the Journal of the Academy of Sciences. As soon as Lame left the podium, Cauchy, another mathematician, also asked to speak.He announced that he had been working on a method similar to Lame's and was about to publish a full proof. Three weeks later, the two each declared that they had deposited at the Academy of Sciences sealed envelopes containing the methods of proof they were eager to claim as their own.Many in the mathematics community secretly hoped that Lame would win the contest rather than Cauchy, a self-righteous fellow, a cultist, and especially unpopular with his colleagues. Unexpectedly, a month later, the German mathematician Kummer sent a letter to the French Academy of Sciences. According to the few details revealed by Lame and Cauchy, he pointed out the logical error made by both of them. Kummer's letter made Lame deflated, but Cauchy refused to admit defeat. For several weeks, he published consecutive articles in defense, and he did not calm down until the end of the summer. Ten years later, the unpopular Cauchy, always self-righteous, submitted his final report on Fermat's last theorem to the French Academy of Sciences: "Mathematical science should provide the geometers, especially Mr. Kummer, with Congratulated for the work done on their desire to solve the problem. The committee considered that if the competition on this problem were withdrawn the prize should be awarded to Mr. Kummer for his work on complex numbers consisting of roots of unity and integers work, that would be a just and beneficial decision of the Academy." "Fermat's Last Theorem - A Mystery That Puzzled the World's Wise Men for 358 Years", (English) Simon Singh (English) by Simon Singh, translated by Xue Mi and published by Shanghai Translation Publishing House