After announcing the proof of the 350-year-old mathematics conjecture, he found that he might have made a fatal mistake...

Andrew Wiles thinks he has solved an old problem, but the story has just begun...

Proof of shocking the world and a little trouble

In June 1993, at a mathematics conference at the University of Cambridge in England, Andrew Wiles made a series of reports, the title is obscure - "model form, elliptic curve, Galois representation" ( Modular Forms, Elliptic Curves, and Galois Representations). His argumentation process was lengthy and skillful, and it came to an end 20 minutes after the third speech. In order to emphasize the results, he made the last one:

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Fermat's Last Theorem, Fermat's Theorem, is a famous conjecture in the history of mathematics, proposed by the 17th century French lawyer and amateur mathematician Pierre de Fermat, but after 350 years there is still no complete proof. Princeton, a professor at Princeton University, hid in the attic of his home and quietly studied this ancient puzzle for seven years. Now, he wants to announce his proof at the venue.

Note: The modular form is an analytic function that accepts only values ​​from the upper half-plane in the complex plane, and this function becomes a certain group under a group operation of the model group. The type of function equation, and the value calculated by the function will also show a certain growth trend. Modular form theory belongs to the category of number theory. Modular forms also appear in other areas, such as algebraic topology and string theory.

Galois group: In mathematics, especially in abstract algebra theory, the Galois theory, named after the French mathematician Évariste Galois, provides a connection between domain theory and group theory. . Applying Galois theory, some of the problems in domain theory can be reduced to a more simple and understandable group theory problem.

This speech shocked the people present and the world. The next day, the matter was on the front page of the New York Times. Wiles had a good reputation for a time. Clothing retailer Gap invited Wiles to participate in the design of a new pair of jeans, but he was eventually rejected. He was named one of the "25 Most Attractive People of the Year" by People, along with Princess Diana, Michael Jackson and Bill Clinton. The famous contributor Barbara Walters contacted him for a visit, and Wiles replied: "Who is Barbara Walters?"

But the celebration did not last long. Once a certificate has been submitted, it must be carefully examined and verified before it can be recognized. Wiles submitted a 200-page certificate to the world's top mathematics journal, Inventiones Mathematicae. The editor of the journal then distributed the manuscript to six reviewers, one of whom was Nick Katz, a mathematician at Princeton University.

Katz and his French colleague Luc Illusie spent two months scrutinizing every logical part of the responsible part. Whenever they encounter some incomprehensible arguments, Katz will send an email to Wiles, and Wiles will reply to clarify the question. But by the end of August, Wiles’ explanation of a problem could not convince the two reviewers. After further research, Wiles understood that Katz found a flaw in the mathematical logic framework of the paper. At first, a simple fix seems to work. But when Wiles started to fix the defect, the fragments of the logical frame began to fall off.

Wiles realized that this was not just a simple mistake, it might even go beyond a repairable defect, and he became more and more scared. If it is a crack, an unrepairable defect, it will make the proof of the whole theorem collapse.

Andrew Wiles, a professor of mathematics at Princeton University. Wiles lasted for seven years and proved the unresolved Fermat's theorem in 350 years. Image source: Associated Press Charles Rex Arbogast

a bleak page in the history of mathematics

Fermat's theorem has incredible simplicity, which was proposed by Fermat in 1637. At the time he was reading Arithmetica, compiled by the ancient Greek mathematician Diophantus. There is a discussion of Pythagorean theorem (Pythagorean theorem) in the book. As we have learned, the square of the oblique side of a right triangle is the sum of the squares of the two right angle sides. The mathematical form can be expressed as x 2 + y 2 = z 2 , where x, y, and z are the two right-angled sides and the oblique-sided lengths in the triangle, respectively.

Diophantine found some positive integer solutions that satisfy the condition, named it "Pythagorean Array", and proved that there are an infinite number of pairs of Pythagorean arrays (strictly speaking, there are infinitely many prime numbers in each other.) Stock array). The simplest examples are right triangles (3, 4, 5), and (5, 12, 13) and (145, 408, 433).

Fermat then asked: Can you find similar arrays in high dimensions? Or is there an positive integer solution for an equation of the form a 3 + b 3 = c 3 ? So a 4 + b 4 = c 4 ? a 10,007 + b 10,007 = c 10,007 ? Fermat's answer is no. A power higher than the second is divided into two sums of the same power, and there are no a, b, c that satisfy the condition. Fermat wrote a famous sentence on the page of "Arithmetic": "I am sure that I found a wonderful proof of this, but the gap here is too small to write."

He also seems to have never elaborated elsewhere. After Fermat's death, his son Samuel published a new version of "Arithmetic," which included all the notes his father had made in the margins. The mathematical propositions in the notes often do not write down the proof process, leaving an extremely attractive challenge. Within a few years, the reader almost proved all the propositions, except for the proposition of the high-dimensional Pythagorean array, which is the "last" proposition that could not be proved (Ferma theorem is also called "Ferma's last theorem") .

For centuries, Fermat's theorem has become the object of pursuit by scientists, amateurs, and "civilians." Every time it seems close, it finds no way out. (The mathematics prince Gauss is one of the early mathematicians who resisted its charm. He refutes it as "an isolated proposition that leaves me with no interest.") The French Academy of Sciences has set up a considerable amount of bounty for it. But even the best mathematicians have made limited progress on Fermat's theorem. Fermat gave the proof of theorem for n=4 in the notes. The mathematician later proved that the great theorem is true for n<100 and the multiple of the corresponding n, but for all n, they can neither prove nor falsify.

However, their exploration brought other gains. From the ruins of failure, a deep theory that opened up a new field of broad mathematics was born. In 1847, the French mathematician Augustin Louis Cauchy and his rival Gabriel Lamé believed that they proved the Fermat's theorem by means of a complex system containing imaginary numbers. A typical imaginary form is bi, where i = √(-1); b is a real number. A plural is written a+bi with a real part a and an imaginary part bi.

Both Cauchy and Lamé's proofs are based on a conjecture that is generally considered correct: complex numbers, like real numbers, can be broken down into a unique set of prime numbers. For example, 6=2×3, except that the order of the changes is broken down into 3×2, there will be no other results. But what bothers are embarrassed is that their contemporary German mathematician Ernst Kummer proved that some complex numbers can be decomposed in multiple ways. For example, 6 + 0i can be decomposed into 2 x (√2 + i) x (√2 – i) or (1 + √5i) x (1 – √5i).

In order to repair the unique prime factor decomposition properties of complex numbers, Kummer created a new concept of algebra—ideal, which plays an important role in the development of later generations of abstract algebra. American mathematician Leonard Eugene Dickson wrote in the early 20th century that Kummer's creation was "one of the most important scientific achievements of the last century . "

But the proof of Fermat's theorem is stuck here. In the middle and late 19th century, most mainstream mathematicians followed Goss's choice and set aside the proof of Fermat's theorem. They have exhausted all possible methods and can't come up with new solutions. Trying to prove the Great Theorem or overthrow it is a huge adventure in time and energy. A promising mathematician may have exhausted his life, pondering meditation, and finally his thoughts are exhausted but he can't produce and work hard.

Cut the entrance - Gushan - Shimura guess

When Wiles was 10 years old, he first came across Fermat's theorem. Like many children with math dreams, he dreams of solving it. But during his Ph.D. in Cambridge, Wiles followed the advice of the mentor and chose to avoid the possible dead end. On the other hand, he turned to learning elliptic curves that are very useful in cryptography. The elliptical curve looks like the surface of a donut.

After Wiles went to the Department of Mathematics at Princeton University, Ken Ribet, a number theorist at the University of California at Berkeley in 1986, presented an unexpected idea that proved far-reaching to the proof of Fermat's theorem. In the past 30 years, two young scholars at the University of Tokyo, Utaka Taniyama and Goro Shimur, proposed the “Gushan-Shicun conjecture” and established elliptic curves (objects of algebraic geometry) and modular forms (used in number theory). An important link between some kind of periodic pure function).

The modular form is a commonly used tool in number theory, which exists in a four-dimensional hyperbolic space (a curved space) with surprising symmetry. Just as a square rotates a quarter turn around the center to coincide with itself; likewise, a form can still coincide with itself after being rotated, reflected, or otherwise transformed. On the other hand, the elliptic curve itself is an algebraic structure. When you draw an image in the complex plane that satisfies y 2 = x 3 + Ax + B (A and B are constants), you can get an elliptic curve.

Gushan and Zhicun proposed a bold and radical idea: the modular form is another form of elliptic curve. If they are right, then all the research results about the modular form can be expressed in the language of the elliptic curve, and vice versa. Proving that this conjecture will be the key to unifying different branches of mathematics.

This may be a breakthrough in proving or falsifying Fermat's theorem. In the 1970s, a French doctoral student named Yves Hellegouarch proved that if Fermat's theorem is wrong, a set of integer solutions of the equation a n + b n = c n (n>2) can be found ( a, b, c), then an elliptic curve satisfying the condition of y 2 = x(x - a n )(x + b n ) can be obtained. Ten years later, German mathematician Gerhard Frey further pointed out that only the above-mentioned elliptic curve would exist if Gushan-Shimura had a wrong guess. Or to be more direct: Once the Gushan-Shimura conjecture is established, the Fermat's theorem will surely be established.

Ribet proved that Frey's conjecture was correct. This made Wiles feel encouraged, and now he can regain the dream of proof of Fermat's theorem without having to deviate from the current mainstream mathematics. He hid in the top floor of his home, determined to prove the Gushan-Shimura guess.

Almost missed

By December 1993, six months had passed since the Cambridge lecture, and Wiles had barely told anyone that the mathematics community had waited for centuries to prove that he was crumbling behind him. Only the reviewer of the paper and his close friend know that there is a flaw. And Wiles did not prove that the rumors of Fermat's theorem began to spread, and mathematicians asked him to publish the manuscript. If there is an error, the peers hope that someone can magically see and fix these defects.

But Wiles is not prepared to let others easily take this honor. He returned to the attic and returned to a state of loneliness, and even Ribet, who had been the unofficial news contact for Wiles, could not reach him. Peter Sarnak, a professor of mathematics at Princeton and a friend of Wiles, said: "I don't know how, people's thoughts are 'you have to prove Fermat's theorem. If you don't prove it, you have trouble.'"

Sarnak persuaded Wiles to find a collaborator to fix the flaw, even if it only "allowed his ideas to be separated from those who are too familiar." Wiles called his former student Richard Taylor. Taylor was already a famous number theorist at Cambridge University. At first, they tried what Taylor called "localization": minor modifications to the methods used in Wiles's incomplete proof to correct errors.

But this does not help. Taylor recalled that they then decided to "expand the scope and spread a larger network to find other ways." Throughout the spring and summer, they have been working, and often even talked over the phone for a long time in the middle of the night. Taylor said: "I have never received such an expensive phone bill."

But by September 1994, their efforts still had no progress. Before preparing to admit defeat to the world, Wiles decided to "finally check" the original method structure, trying to find out exactly why it could not work. In the BBC documentary The Proof, he tells the story. “Suddenly, completely unexpected, I had an incredible discovery.” In the embers of the failed technology , there is precisely a tool to prove another conjecture. That tool is the " Iwasawa theory " (Iwasawa theory). He gave up this method three years ago, but now he can use it to completely compensate for the defects, thus proving the Fermat's theorem. "It's so beautiful, it's so simple and elegant. I stare at it, it's unbelievable."

With this theory, Wiles and Taylor quickly fixed the loopholes in the paper within a few weeks. In May 1995, they published two papers on all the work in the world's leading journal, Annals of Mathematics. The final proof and accompanying discussion is 130 pages long.

Is this proof that Fermat did not write? That is, the "famous proof" that "The Arithmetic" page is too narrow to write? The only reasonable answer is "NO". In order to prove Fermat's theorem, Wiles used the latest mathematical tools and ideas, which were born far later than Fermat's era. Most mathematicians believe that Fermat's theorem is summed up in error. If he is convinced that he knows the method of proof, it is likely to just confuse himself.

But what matters is not Fermi's personal right and wrong. The ancient Greeks ignited the source of the field of number theory, and Fermat's misleading boasting turned the dying flame into a major branch of mathematics. The mathematical legacy of his imperfect genius is far more important than the trivial matter of how he came to guess.

For Wiles, fortunately his mistake is only a small defect that can be made up.