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This is a mystery of challenging the wise. I don't know how many people spent their whole lives for it, but finally fell at its feet.

Its proposition is extremely simple and fascinating, but it is difficult and daunting to crack. It challenged people for 358 years and was finally deciphered by the British mathematician Andrew Wells. On September 19, 1994, Wells solved the mystery of Fermat's Great Theorem and became the first person to reach the pinnacle of mathematics.

It was Pierre de Fermat who presented the problem to the world by Andrew Wells. Born in France in 1601, Fermat worked as a lawyer and later became a local judicial magistrate, but did his own "private work" in his spare time, making achievements in number theory, geometry and mathematical analysis.

In 1934, it was found that Newton's manuscript mentioned that his invention of calculus was inspired by Fermat tangent.

Pierre de Fermat was called the "Prince of Amateur Mathematics" at that time, and he liked it very much. He spends all his spare time on mathematics, purely out of curiosity and amusement, but never cares about the use of mathematics. Whenever he came up with a strange proposition for himself and racked his brains to get a strange solution, he would make fun of his friends with the problem. When his friends were puzzled, he was secretly smug with delight. Fermat thought there was nothing wrong with him. Descartes called him a "bragging king", while the English mathematician John Wallis called him a "fucking Frenchman".

For a while, Fermat studied Diophantu's book, and in the second episode of the arithmetic Theorem, he read the Pythagorean Theorem, which aroused his interest. The Pythagorean theorem in China is similar to the Pythagorean theorem. Pythagorean Theorem comes from the dialogue between Zhou Gong and Shang Gao, that is, "Pythagorean Theorem". Although it is 500 years earlier than Pythagorean Theorem, it is only a special case of right triangle Pythagorean relationship. Pythagoras not only proved that this Pythagorean relation is applicable to all right triangles, but also extended it beyond right triangles according to logical reasoning.

Pythagoras found that there can be an infinite number of right triangles whose sides are natural numbers, which shows that the Pythagorean relation of right triangles must mean that there is a general law of natural numbers, from which he established a general theorem of natural numbers. That is, "you must find a natural number, and its square must be equal to the sum of the squares of the other two natural numbers." This is the Pythagorean theorem, which originated from the study of right triangles.

After Fermat saw the theorem, he modified the theorem slightly, that is, replacing the square with the n-th power. He thought, can we find a natural number that is not 0 whose n-th power is equal to the sum of the other two natural numbers? As shown in the following formula:

Obviously, there is no problem when n is 1; when n is 2, it is the Pythagorean theorem; when n > 2, what happens? It is impossible to know what Fermat did later, but Fermat left these words on the page of his book, "when n is greater than 2, it is impossible, and here I am sure I have found a wonderful proof, but the space here is too small to write." Fermat left the riddle to posterity.

Fermat's great theorem, that is, when n is greater than 2, it is impossible to find a natural number whose n power is equal to the sum of the other two natural numbers. People can't help but ask, what kind of wonderful proof did Fermat give? Did Fermat give this wonderful proof? It is said that a hundred years later, the great mathematician Euler sent someone to Fermat's former home, hoping to find the remaining manuscript, but he probably didn't find anything.

Fermat's Great Theorem is an unforgettable mystery of the century. It is like a "curse" that firmly binds people to escape. A hundred years after Fermat, Euler first made a breakthrough when he proved that there were no natural solutions for nasty 3 and nasty 4. After Euler, Fermat's Great Theorem stopped here, and no obvious progress was made even if someone asked about it.

Nearly a hundred years later, there suddenly appeared a strange woman, Maria Germain Sophie.

Maria Germain Sophie's appearance is a miracle of miracles. It was the time of the French Revolution, and it was a miracle for women to learn science. Sophie's interest in mathematics was moved by Archimedes. When the Roman soldiers broke into the house, Archimedes lost his life because he devoted himself too much to his research and did not hear the shouting. The story moved Sophie and introduced her to the field of science.

After entering the Paris Comprehensive Institute of Technology, Sophie can only remain anonymous, she used the identity of "Mr. LeBlanc". It turned out that the LeBlanc had dropped out of college, and Sophie's impostor went unnoticed. But her homework gave her away. Her homework showed an extraordinary talent for mathematics, which shocked her teacher Lagrange. When Lagrange wanted to make an appointment with Mr. LeBlanc for an interview, Sophie told the truth. Instead of being angry, Lagrange admired the strange woman's talent and tenacity, and from then on he became Sophie's mentor and friend.

Sophie spent several years studying Fermat's great theorem. She found an innovative way to prove that when n equals a prime, there is almost no solution to the equation less than 100. Later, Dillich and Legendre independently proved the case of nasty 5 by using Sophie's method.

Over the years, Sophie has been sending her research results to the mathematician Gauss, and when she wrote, she called herself "Mr. LeBlanc". Just then when Napoleon attacked Germany, Sophie suddenly thought of Archimedes. She felt uneasy for Gauss and feared that the same misfortune would befall him. Among the French generals who led troops to attack Prussia, there was a friend of Sophie, General Paniti, who told him to pay attention to the safety of Professor Gauss. When Gauss learned that Mr. LeBlanc was a woman, he was so surprised that he wrote a long letter to Sophie, "I can't express my surprise and admiration in words." how hard to believe that the honorable Mr. LeBlanc is such an outstanding woman! "

In this way, the mathematical heroine was seen through by Lagrange and Gauss, two great figures in the field of mathematics at that time, in a completely unexpected way. They knew each other, knew each other, and became friends.

Another hundred years have passed since Sophie. In the meantime, although the French Academy of Sciences and the University of Gottingen in Germany issued notices respectively, many of the crackers of Fermat's great theorem were eager to try, and most of them ended in vain.

It wasn't until after World War II that two Japanese youths appeared. They were Taniyama and Shimura.

Tanayama, these are two friends with different personalities. Tanayama is brightly dressed, with poet-like high-pitched and informal details, while Shimura is serious and peaceful.

In the beacon of Shimura, the Japanese mathematical circle was isolated from the rest of the world. Taniyama and Shimura not only studied the outdated "modular form" theory at that time, but also studied the natural number solutions of elliptic equations in algebraic number theory.

It was these two topics that led them to make a wonderful discovery. In a series of calculations, they found a law that each elliptic equation corresponds to a modular form. But they are not sure whether this is a universal law, so they put forward a conjecture that "every elliptic equation has a modular form" and think that it is a universal law.

At that time, this result was not optimistic, because the modular form of the elliptic equation was completely irrelevant to the general modular form. However, in the more than ten years since they put forward this conjecture, some special cases have been confirmed. Gradually, the "Gushan-Zhicun conjecture" has become a subject of concern in the field of mathematics, and some people expect that this subject may become the beginning of a new branch of mathematics. What is even more unexpected is that the "Gushan-Zhicun conjecture" has become the key to cracking Fermat's great theorem.

In 1958, Shimura went to Princeton University as a visiting scholar, but Tanayama committed suicide in Japan. The journey to crack Fermat's great theorem came to a standstill.

Like a plan, Gushan committed suicide in the 1950s, while another math geek, Wells, was born in the 1950s.

Wells was born in the home of Oxford professors in Cambridge, England. The edification of two famous schools made him like reading and math more since he was a child. At the age of 10, he read Fermat's Great Theorem in the library. Such a simple formula has not been cracked for more than 300 years. He tried to find proof, although it was futile, but the proof of Fermat's great theorem became a major event that haunted him all the time.

After graduating from college, Wells began to study pure mathematics while teaching at the university. He now understands that the simpler things seem, the deeper the trap will be, and that it is dangerous to devote energy to the puzzle of Fermat's Great Theorem. Mathematics, on the other hand, can lead to a lifetime of exhaustion, but the end result will be zero. To deal with the difficult problems of the century, we need solid skills, extraordinary skills, strong will, strict logical thinking, extraordinary intuition and wisdom, and preparations for fruitless results in the end.

Of course, there has to be some kind of luck. It seems that Wells was born lucky. He majored in elliptic curves at Cambridge. Invisible, the elliptical curve builds a bridge for him to "Tanayama-Shimura conjecture".

It was at this time that he was again blessed by fate. In 1984, a symposium of mathematicians was held in Germany. At the meeting, a German mathematician, Farley, proposed a workaround to prove Fermat's great theorem. He linked Fermat's great theorem to the elliptic equation, and he believed that if the Tanayama-Shimura conjecture was generally correct, the corresponding module form of the elliptic equation would become "incredibly strange" and even impossible to exist. If this is the case, once the "Tanayama-Shimura conjecture" is established, Fermat's Great Theorem will be proved by using the method of absurdity.

In 1986, Farley's reasoning was further advanced by Kenny Liebert of the University of California, Berkeley. He confirmed that the modular form mentioned by Farley really did not exist. In this way, the proof of Fermat's great theorem is naturally deduced into such a result that as long as it is proved that the "Gushan-Zhicun conjecture" is true, Fermat's great theorem will be cracked.

Wells is secretly determined that he has only one goal, that is, to find the proof of the Tanayama-Shimura conjecture. He excluded all trifles, cut off all interpersonal communication, and no longer participated in any activities except for the necessary teaching and seminars. He devoted all his energies to the "Tanayama-Shimura conjecture".

Since Wells lived in seclusion in 1986, he has lived a very lonely life with pen and paper as if he were groping in the dark. As he said, "it's like stepping into a dark building. The first room was so dark that you stumbled on the furniture and slowly figured out the location of each furniture. Six months later, I finally found the light switch and lit up the whole room at once. Then I stepped into another room and stayed in the dark for another six months. In this way, each breakthrough may only take a day or two, but without the exploration of the first six months, such a breakthrough would not have happened at all. "

Wells' research was not plain sailing. He adopted mathematical induction in the first three years. Three years later, between 1990 and 1991, he ran up against a brick wall and finally found that it was a dead end. When he was in trouble, he spent a long time alone in calculation, which made him exhausted.

Wells, who is doing research, is at a loss. He wants to take a break from change, so he walks out of isolation and goes to Boston to listen to the latest research from his peers. It did not occur to him that the predicament he had encountered was the darkness before the dawn, and the dawn was right in front of him. He suddenly came across an article written by Fareh, which seemed to be written for him. Inspired by this article, he changed course and abandoned the previous Iwazawa theory and began to try to perfect Colliagin-Fareh's theory. This change made him progress so fast that one day in May 1993, he said to his wife, I have solved Fermat's Great Theorem.

Wells finally showed up, and in June 1993, he unveiled his work at the Mathematical Society of Cambridge University. The lecture is divided into three sessions, namely, modular form, elliptic curve and Galois representation theory. Although he did not specify the connection with the "Tanayama-Shimura conjecture", by the end of the second lecture, the mathematical community had spread Wells' important findings. When it came to the third lecture, it was June 23, 1993, and almost all the mathematics colleagues from Oxford and Cambridge came. They packed the meeting room, and everyone was very excited about this "lecture of the century". Wells' speech was wonderful and ended with thunderous applause.

On the second day of the lecture, Andrew Wells cracked Fermat's Theorem in newspapers around the world, and his name made headlines. Time magazine called him "one of the most intriguing people in the world" in this year's person of the year.

Wells was immersed in happiness, but he never expected that there was a small mistake in his 200-page manuscript, which was an omission in the reasoning of the Colliagin-Fareh theory.

Wells doesn't have the pleasure of studying alone for the last seven years, and he will work under the watchful eye of dozens, hundreds or even thousands of people. He said: "Learning in full view of the public is really not what I expect. I do not like it very much." Six months have passed since that "glorious moment", and his paper has not yet been made public, and the mathematical community is already whispering that there is something wrong with his proof. On December 4, 1993, he had to admit that there was a loophole in his certificate and was trying to correct it.

In the summer of 1994, he was on the verge of despair. He thought repeatedly that if he gave up, even if there were omissions in the proof close to the "Tanayama-Shimura conjecture", his ideas and work could still be called first-class and a successful retirement, but he did not want to just admit defeat. On September 19, 1994, the dawn finally appeared again.

Recalling the incident, he said: "on the morning of September 19, I sat at my desk and examined the Colliagin-Fareh theory. I don't want it to work at all. I just want to know why it doesn't work. All of a sudden, I had an idea that if I combined the previously abandoned Iwazawa theory with the Koriagin-Fareh theory, it would be enough to prove the Tanayama-Shimura conjecture! I stared at it for 20 minutes and couldn't believe I had been ignoring it. That day, after a while, I walked down the hallway of the math department and went back to the office to see if it was still there. It's still here! I can't help myself. I'm so excited. This is the most important moment in my life. Nothing I do, whether in the past or in the future, is not as meaningful to me as this moment. "

Wells realized his dream, and on October 24, 1995, his results were finally published in the Mathematical Yearbook under the title "pattern Elliptic Curve and Fermat's Great Theorem". The manuscript was 150 pages long and took seven years.

Wells stood in front of the statue of Fermat, and the spell of the century that challenged human wisdom for 358 years was finally completely broken. This is the pinnacle of the study of modern geometric algebra and number theory, and it can be called the achievement of the century. For this reason, Wells won the title of knighthood.

Source: 365th days in the History of Science author: Wei Fengwen Wu Yi part of the online copyright belongs to the original author this article comes from Wechat official account: Origin Reading (ID:tupydread), author: Wei Fengwen, Wu Yi, Editor: Zhang Runxin

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