Modern mathematics is like a towering tree, flourishing. There are countless math problems hanging on this vigorous tree. Among them, the four-color map problem, Fermat's last theorem and Goldbach's conjecture are the most dazzling. They are called the three major mathematical problems in modern times. For more than 300 years, Fermat's Last Theorem has made many famous mathematicians in the world exhausted their efforts, and some even spent their whole lives. The mysterious veil of Fermat's Last Theorem was finally unveiled at 1995, which was proved by 43-year-old British mathematician Wells. This is considered as "the greatest mathematical achievement of the 20th century". ? The origin story of Fermat's last theorem involves two mathematicians separated by 1400 years, one is Diophantine in ancient Greece and the other is Fermat in France. Diophantine activities took place around 250 AD. 1637, Fermat in his thirties was reading the French translation of Diophantine's masterpiece Arithmetic. In his book, he talked about indefinite equations. x2+？ y2？ =z2？ In the blank of this page, it is written in Latin: "The cube of any number cannot be divided by the sum of the cubes of two numbers; The fourth power of any number cannot be divided by the sum of the fourth powers of two numbers. Generally speaking, it is impossible to divide a power higher than a quadratic power by the sum of two powers of the same power. I found a wonderful proof of this judgment, but unfortunately the space here is too small to write down. " ? After Fermat's death, people found this passage written on their eyebrows when sorting out his relics. 1670 His son published Fermat's pagination note, and everyone knows this problem. Later, people called this assertion Fermat's Last Theorem. Expressed in mathematical language is: looks like xn? +yn？ =zn？ When n is greater than 2, there is no positive integer solution. Fermat is an amateur mathematician and is known as the "king of amateur mathematicians". 160 1 year, he was born in a leather merchant's family near Toulouse in southern France. I was educated at home when I was a child. When he grew up, his father sent him to university to study law and became a lawyer after graduation. Member of Toulouse City Council, tel. 1648. He loves mathematics very much and spends all his spare time studying mathematics and physics. Because of his quick thinking, strong memory and indomitable spirit necessary for studying mathematics, he has achieved fruitful results, making him one of the great mathematicians in the17th century. Difficult exploration? At first, mathematicians tried to rediscover Fermat's "wonderful proof", but no one succeeded. Euler, a famous mathematician, proved the equation through infinite deduction? x3+？ y3？ =z3？ And then what? x4？ +? y4？ =z4？ There can be no positive integer solution. Because any integer greater than 2, if not a multiple of 4, must be an odd prime number or a multiple of it. So as long as it can be proved that n = 4 and n is an arbitrary odd prime number, the equation has no positive integer solution, and Fermat's last theorem is completely proved. The case of n = 4 has been proved, so the problem focuses on proving that n is equal to an odd prime number. Proved by Euler? n=？ 3,? n=？ 4、？ 1823 and? Legendre and Dirichlet independently proved in 1826? n=？ 5. The situation of 1839 has the cripple proved it? n=？ The situation of 7. In this way, the long March to prove odd prime numbers began. Among them, the German mathematician Cuomo made an important contribution. He introduced the concepts of "ideal number" and "cyclotomic number" invented by himself with the method of modern algebra, and pointed out that Fermat's Last Theorem may be incorrect only when n is equal to some values called irregular prime numbers, so we only need to study these numbers. Such figures, within 100, are only 37,59,67. He also specifically proved when? n=？ 37, 59, 67, equation xn+? Yn = Zn cannot have a positive integer solution. This pushes Fermat's last theorem to the point where n is within 100. Kummer's batch proof theorem is regarded as a major breakthrough. 1857 won the gold medal of the Paris Academy of Sciences. This "long March" proof method, although constantly refreshing records, such as? 1992 goes further to n = 1000000, but this does not mean that the theorem is proved. It seems that we need to find another way. Who was awarded the 65438+ Million Mark Award? Since Fermat's time, the Paris Academy of Sciences has twice offered medals and prizes to those who proved Fermat's Last Theorem, and the Brussels Academy of Sciences has also offered a large prize, but all of them have no results. 1908, when Wolfskeil, a German mathematician, died, he presented his 65,438+ten thousand marks as a prize for solving Fermat's last theorem to the Gottingen Scientific Society in Germany. The Goettingen Science Society announced that the prize is valid for 100 years. The G? ttingen Scientific Association is not responsible for reviewing manuscripts. 65438+ million marks was a lot of wealth at that time, and Fermat's last theorem was a problem that all primary school students could understand. Therefore, not only people who specialize in mathematics, but also many engineers, priests, teachers, students, bank employees, government officials and ordinary citizens are studying this problem. In a short time, there are thousands of certificates issued by various publications. ? At that time, there was a magazine in Germany called "Records of Mathematical Literature", which volunteered to identify papers in this field. By the beginning of 19 1 1, * * has checked11proofs, all of which are wrong. Later, I really couldn't stand the heavy burden of peer review and announced that I would stop this review and appraisal work. However, the wave of proof is still surging. Although the German currency depreciated many times after the two world wars, it is of little value to exchange the original 654.38+ million marks for later marks. However, the precious spirit of loving science still encourages many people to continue this work. The belated proof has made a lot of achievements in the proof of Fermat's last theorem through the efforts of predecessors, but there is no doubt that there is still a long way to go to prove the distance theorem. What shall we do? In the future, we must use a new method. Some mathematicians have already used the traditional method-transformation problem. People associate the solution of Diophantine equation with a point on algebraic curve, which becomes a transformation of algebraic geometry, and Fermat problem is only a special case of Diophantine equation. On the basis of Riemann's work, 1922, the British mathematician Mo Deer put forward an important conjecture. Let F(x, y) be a rational coefficient polynomial of two variables x, y, then when the curve F(x, y)=? When the genus of 0 (a quantity related to a curve) is greater than 1, the equation F(x, y) = 0 has at most a finite set of rational numbers. 1983, the 29-year-old German mathematician Fortins proved Mo Deer's conjecture in the Soviet Union with a series of results of Shafara's algebra and geometry. This is another major breakthrough in the proof of Fermat's last theorem. Fielding won the Fields Prize with 1986. Wells still uses algebraic geometry to climb. He connected other people's achievements wonderfully, learned from the conquerors who walked this road, and noticed a brand-new detour path: if the Taniyama-Zhicun conjecture holds, then Fermat's Last Theorem must hold. This was discovered in 1988 by German mathematician Ferrer when he studied a conjecture of Japanese mathematician intellectual village about 1955 elliptic function. Wells was born into a theological family in Oxford, England. He was very curious and interested in Fermat's last theorem since he was a child. This wonderful theorem led him into the palace of mathematics. After graduating from college, he began his childhood fantasy and decided to fulfill his childhood dream. He studied Fermat's last theorem in great secrecy and kept his mouth shut. After seven years of poverty, I persisted until June 23 1993. On this day, a regular academic report meeting is being held in the hall of Newton Institute of Mathematics at Cambridge University. The speaker Wells gave a two-and-a-half-hour speech on his research results. 10: 30, at the end of the report, he calmly announced: "Therefore, I proved Fermat's last theorem". This sentence, like a thunder, set many hands that only need to do routine work in the air, and the hall was silent. After half a minute, thunderous applause seemed to overturn the roof of the hall. British scholars are ecstatic regardless of their elegant gentlemanly manners. The news quickly caused a sensation all over the world. Various mass media have reported it and called it "the achievement of the century". It is believed that Wells finally proved Fermat's Last Theorem and was listed as one of the top ten scientific and technological achievements in the world in 1993. But soon, the media quickly reported an "explosive" news: Wells' 200-page paper was found to be flawed when it was submitted for trial. In the face of setbacks, Wells did not stop. It took him more than a year to revise the paper and correct the loopholes. At this time, he was "haggard for Iraq", but he "gradually widened his skirt and never regretted it." 1September 1994, he wrote a new paper with a page of 108 and sent it to the United States. The paper passed the examination successfully, and the Yearbook of Mathematical Journal published his paper in May 1995. Wells obtained the wolf prize in mathematics of 1995 ~ 1996. After that? After more than 300 years of continuous struggle and the efforts of several generations, mathematicians have made many important discoveries around Fermat's last theorem, which has promoted the development of some branches of mathematics, especially the progress of algebraic number theory. The core concept "ideal number" in modern algebraic number theory is put forward to solve Fermat's last theorem. No wonder Hilbert, a great mathematician, praised Fermat's Last Theorem as "a hen that lays golden eggs".

Proof Lemma of Fermat's Last Theorem: kn+ 1 is not a perfect power number (k is a positive rational number, n is a natural number, and n > 2). ? Proof: 1. When k is a positive integer, let kn+ 1=an then k < a < k+ 1? ∴a must be a mixed decimal (in fact, A is an irrational number)? ∫ A mixed decimal to the nth power or a mixed decimal? ∴an is a mixed decimal. And ∵kn is a positive integer,? ∴an？ –? Is kn a mixed decimal or a pure decimal? ∴an–kn≠ 1？ So an≠kn+ 1? Second, when k is a false score, let kn+ 1=an then k < a < k+ 1? ∵a≠k？ ∴? A and k have the following two situations:? ( 1)? The integer part is the same, but the decimal part is different; ? (2) The integer part is different and the decimal part is different. In the above two cases, the fractional parts of an and kn will not be the same. ∴ an–kn is a mixed decimal or a pure decimal, that is, an–kn ≠1,then kn+ 1=an? Third, when? When k is a true fraction, let kn+ 1 = c, and let an=c, then kn+ 1=an, 1 < a < k+ 1? ∵k≠a？ There is only one case of k and a, where the integer part is different and the decimal part is different? The fractional parts of an and kn are not equal. Must it be mixed decimal or pure decimal? ∴an– 1？ So what? an≠kn+ 1？ Based on the above three situations, we can know that n√? kn+？ 1 is an irrational number, that is, when n≥2, kn+ 1 is not a complete power number. √? (In fact, kn- 1 is not a perfect force)? Fermat's last theorem proposition: an+bn≠cn(a, b, c are natural numbers, n > 2)? Proof: ∫an+bn =? an〔b/a〕n？ +? 1? 〕? Know whether (b/a)n is reasonable? +? 1 is not a complete power of n? ∴? an〔b/a〕n？ + 1? 〕? Incomplete &; # 178; Power? So an+bn≠cn? So Fermat's last theorem holds? Note: it can also be expressed as: cn-an=? an〔c/a〕n？ -? 1? 〕