When the integer n > 2, the indefinite equation about x, y and z.

x^n + y^n = z^n.

The integer solution of is trivial, that is,

When n is an even number: (0, m, m) or (m, 0, m)

When n is odd: (0, m, m) or (m, 0, m) or (m, -m, 0)

This theorem was originally called Fermat's conjecture, which was put forward by the French mathematician Fermat in the17th century. Fermat claimed that he had found a wonderful piece of evidence. But after three and a half centuries' efforts, this century's number theory problem was successfully proved by British mathematician andrew wiles and his student richard taylor at Princeton University on 1995. It is doubtful whether Fermat really found the correct proof because many new mathematics are used in the proof, including elliptic curves and modular forms in algebraic geometry, Galois theory and Heck algebra. Andrew wiles won the Fields Prize of 1998 and the Shaw Prize for Mathematics in 2005 because he successfully proved this theorem.

Edit this study history.

1637, when Fermat was reading the Latin translation of Diophantine arithmetic, he wrote beside the eighth proposition in Volume 1 1: "It is impossible to divide a cubic number by the sum of two cubic numbers, it is impossible to divide a quartic power by the sum of two quartic powers, and it is even more impossible to divide a power higher than quadratic by the sum of two powers of the same order. In this regard, I am sure that I have found a wonderful proof, but unfortunately the space here is too small to write down. " (Latin original: "Cui us rei demonstrates em mirabile m sane de Texi. Hanks deposit is very small, not Caperet. " After all, Fermat didn't write a proof, and his other conjectures made great contributions to mathematics, which inspired many mathematicians' interest in this conjecture. The related work of mathematicians enriches the content of number theory and promotes its development.

For many different N's, Fermat's Last Theorem has already been proved. But mathematicians are still confused about the general situation of the first 200 years.

1908, Wolfsk, Germany announced that it would award 65438+ million marks to the first person who proved the theorem within 100 years after his death, which attracted many people to try and submit their "proofs". After World War I, the mark depreciated sharply, and the charm of this theorem also declined greatly.

1983, en: gerd faltings proved Mo Deer's conjecture, and it was concluded that when N >: 2 (n is an integer), coprime A, B and C have only finite sets, which makes an+bn = cn.

1986, gerhardt Frey put forward "ε-conjecture": if a, b, c, a n+b n = c n, that is, if Fermat's last theorem is wrong, the elliptic curve y 2 = x (x-a n) (x+b). Frey's guess was immediately confirmed by Kenneth Rebett. This conjecture shows the close relationship between Fermat's Last Theorem and elliptic curve and module form.

1995 wiles and Taylor proved the Taniyama-Zhicun conjecture in a special case, and the Frey elliptic curve is just in this special case, thus proving Fermat's last theorem.

Wiles's process of proving Fermat's Last Theorem is also very dramatic. It took him seven years to obtain most of the evidence without being known. Then in June of 1993, he published his certificate at an academic conference, which immediately became the headlines of the world. But in the process of examining and approving the certificate, the experts found a very serious mistake. Wiles and Taylor then spent nearly a year trying to remedy it, and finally succeeded in a method abandoned by wiles in September 1994. This part of the proof is related to Iwasawa's theory. Their proof was published in 1995 Mathematics (en: Mathematics Yearbook).

1: Euler proved the case of n=3 by the unique factorization theorem.

2. Fermat himself proved that n=4.

3: 1825, Dirichlet and Legendre proved the case of n=5, using the extension of Euler method, but avoiding the unique factorization theorem.

4: 1839, French mathematician Lame proved the case of n=7. His proof uses a clever second tool closely combined with 7 itself, but it is difficult to generalize to the case of n= 1 1. Then, in 1847, he proposed the method of "cyclotomic integer" to prove it, but it didn't succeed.

5.kummer put forward the concept of "ideal number" in 1844, and he proved that Fermat's Last Theorem holds for all prime index n less than 100, and this research has reached a stage.

6. Leberg submitted a certificate, but it was rejected because there were loopholes.

7. Hilbert has also studied it, but there is no progress.

8: 1983 the german mathematician faltings proved an important conjecture-modal conjecture: the equation of square of x+square of y = 1 can be understood at most by a limited number. He won the Fields Medal for this.

At 9: 1955, Japanese mathematician Yutaka Taniyama first guessed that there was some connection between elliptic curve and another curve that mathematicians knew better-modulus curve. Gu Shan's conjecture was further refined by Wei Yi and tangcun Goro, forming the so-called "Gu Shan-tangcun conjecture". This conjecture shows that elliptic curves in rational number domain are all modular curves. This abstract conjecture confuses some scholars, but it makes the proof of Fermat's Last Theorem a step forward.

10: 1985, German mathematician Frey pointed out the relationship between "Gu Shan-intellectual village conjecture" and "Fermat's last theorem"; He put forward a proposition: if Fermat's last theorem is n > true, that is, there are a set of non-zero integers A, B and C, so that the n power of A+the n power of B = the n power of C (n >: 2), then the elliptic curve constructed by this set of numbers with the shape of Y squared = X (the n power of X +a) multiplied by (the n power of x-B) cannot be a modular curve. Despite his efforts, his proposition contradicts the "Gushan-Zhicun conjecture". If we can prove these two propositions at the same time, we can know that Fermat's Last Theorem is not based on reduction to absurdity, and this assumption is wrong, thus proving Fermat's Last Theorem. But at that time, he did not strictly prove his proposition.

1 1: 1986, American mathematician Bert proved Frey's proposition, so he wanted to focus on the "Gushan-Zhicun conjecture".

12:1In June, 1993, British mathematician Wells proved the "Taniyama-Zhicun conjecture" for a large class of elliptic curves in the rational number field. Because he showed in his report that Frey curve belongs to this elliptic curve, it also shows that he finally proved Fermat's last theorem. However, experts found loopholes in his proof, so Wells proved Fermat's Last Theorem completely and satisfactorily in September 1994 after more than a year's efforts.

Edit this paragraph to prove the process.

1676, mathematicians proved that n = 4 by infinite descent method according to several hints of Fermat. German mathematician Leibniz and Swiss mathematician Euler also proved that n = 4 in 1678 and 1738 respectively. 1770 Euler proved that n = 3. 1823 and 1825, French mathematician Legendre and German mathematician Dirichlet successively proved that n = 5. 1832 Dirichlet tried to prove n = 7, but only proved n = 14. 1839, the French mathematician Lame proved that n = 7, and then it was simplified by the French mathematician Leberg ... 19 century, the greatest contribution was the German mathematician Kumar, who spent more than 20 years from 1844 to establish the ideal number theory, laying the foundation for algebraic number theory; Kumar proved that Fermat's Last Theorem holds when n < 100, except for 37, 59 and 67.

In order to promote the proof of Fermat's Last Theorem, the Academy of Sciences in Brussels and Paris awarded several awards. 1908, German mathematician Wolfskeil offered a reward of 654.38 million marks at the Royal Scientific Society in G? ttingen, and fully considered the difficulty of proof. The deadline was set at 100. Math fans are eager for this and send "proofs" to mathematicians in succession, hoping to win the championship with only a few pages of elementary transformation. German mathematician Landau printed a batch of postcards for students to fill in: "Dear sir or madam, your proof of Fermat's Last Theorem has been received and is now returned. The first error appears in line _ on page _. "

Mathematicians not only use extensive and profound mathematical knowledge in the process of solving problems, but also create many new theories and methods, which have made inestimable contributions to the development of mathematics. 1900, Hilbert put forward 23 unsolved problems, although he did not include Fermat's last theorem, but regarded it as a typical example of constantly generating new theories and methods in solving these problems. Hilbert is said to claim that he can prove it, but he thinks that once the problem is solved, there will be no beneficial by-products. "I should be more careful not to kill this hen that often lays golden eggs for us."

Mathematicians advanced slowly and persistently until 1955 proved that n < 4002. The appearance of large computers speeds up the proof. German mathematician wagstaff proved n < 125000 in 1976, and American mathematician rozelle proved n < 4 100000 in 1985. However, mathematics is a rigorous science. No matter how big the value of n is, it is still limited, and the distance from finite to infinite is long and far away.

1983, the 29-year-old German mathematician Fortings proved the Mo Deer conjecture in algebraic geometry and won the Fields Prize at the 20th International Congress of Mathematicians. This prize is equivalent to the Nobel Prize in mathematics, and it is only awarded to young mathematicians under the age of 40. Mo Deer conjecture has a direct inference: the equation has at most a finite number of integer solutions in the form of x n+y n = z n (n ≥ 4). This is a beneficial breakthrough in the proof of Fermat's last theorem. There is still a big gap from "limited multi-groups" to "one group without", but a big step has been taken from infinite to limited.

1955, the Japanese mathematician Yutaka Taniyama put forward the Taniyama conjecture, which belongs to the category of algebraic geometry. Frey, a German mathematician, pointed out in 1985 that if Fermat's last theorem does not hold, so does Taniyama's conjecture. Then the German mathematician Pell put forward Pell conjecture, which made up for the defects of Frey's view. At this point, if both Taniyama conjecture and Pell conjecture are proved, Fermat's last theorem is self-evident.

A year later, Peter, a mathematician at the University of California, Berkeley, proved Pell's conjecture.

1In June, 993, andrew wiles, a British mathematician and professor at Princeton University, gave a series of academic lectures on algebraic geometry at the Newton Institute of Mathematics at Cambridge University. In his last lecture "Elliptic Curve, Model and Galois Representation" on June 23rd, wiles partially proved the Taniyama conjecture. The so-called partial proof refers to wiles's proof that Taniyama conjecture is valid for semi-stable elliptic curves-thankfully, the elliptic curves related to Fermat's Last Theorem are semi-stable! At this time, more than 60 well-known mathematicians present realized that Fermat's Last Theorem, which had troubled the mathematics field for three and a half centuries, had been proved! After the speech, the news spread like wildfire Many universities hold parades and carnivals. In Chicago, the police even took to the streets to maintain order.

Edit the proof method of this paragraph

In the 1950s, Yutaka Taniyama, a Japanese mathematician, first put forward a conjecture about elliptic curves, which was later developed by Goro Shimamura, another mathematician. At that time, no one thought that this conjecture had anything to do with Fermat's last theorem. In 1980s, German mathematician Frey linked Yutai Taniyama's conjecture with Fermat's last theorem. What andrew wiles did was to prove that one form of Yutai Taniyama's conjecture was correct according to this connection, and then deduced Fermat's last theorem.

This conclusion was officially published by Willis at the seminar of Newton Institute of Mathematics, Cambridge University, USA on June 2 1, 1993. This report immediately shocked the whole mathematics field, and even the public outside the mathematics door paid infinite attention. However, wiles's certificate was immediately found to have some defects, so it took wiles and his students 14 months to correct it. 1September 1994 19 They finally handed over a complete and flawless scheme, and the nightmare of mathematics finally ended. 1In June 1997, wiles won the Wolfskeil Prize at the University of G? ttingen. At that time,100000 counterfeit goods were about $2 million, but when wiles received them, it was only worth about $50000, but andrew wiles has gone down in history and will be immortal.

Expressed by indefinite equation, Fermat's last theorem is: when n >; 2. The indefinite equation x n+y n = z n has no integer solution of xyz≠0. In order to prove this result, we only need to prove the equations x 4+y 4 = z 4, (x, y) = 1 and x p+y p = z p, (x, Y) = (X, Z) = 65438+.

The case of n = 4 has been solved by Leibniz and Euler. Fermat himself proved that p = 3, but the proof was incomplete. Legendre [1823] and Dirichlet [1825] proved the case of p = 5. 1839, Lame proved the case of p = 7. 1847, the German mathematician Cuomo made a breakthrough in Fermat's conjecture. He founded the ideal number theory, which made him prove that when P

Modern mathematicians also use large electronic calculators to explore Fermat's conjecture, which greatly advances the number of p until 1977, when wagstaff proved p; 0, y>0, z>0, n>2, let x n+y n = z n, then x >；; 10 1,800,000。

Description:

Prove Fermat's last theorem is correct

(that is, x n+y n = z n for n > 2 There is no positive integer solution)

It is only necessary to prove that x 4+y 4 = z 4, x p+y p = z p (p is an odd prime number) has no integer solution.