If expressed by indefinite equation, Fermat's last theorem is: when n > 2, the indefinite equation xn+y n = z n has no integer solution of xyz≠0. In order to prove this result, it is only necessary to prove that the equations x4+y 4 = z 4, (x, y) = 1 and xp+yp = zp, (x, y) = (x, z) = 1 [p is an odd prime number] have no XYZ.
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, and xn+y n = z n, then x >；; 10 1,800,000。
It is proved that natural numbers A, B, C do not exist and satisfy A N+B N = C N (n > 2, n∈Z). This is the famous Fermat theorem. The mystery of Fermat's last theorem was finally unveiled in 1995, which was proved by the 43-year-old British mathematician A. wiles. 1637, Fermat wrote next to the eighth proposition of volume 1 1 when reading the Latin translation of Diophantine's arithmetic: "It is impossible to put a It is also impossible to divide a quartic power by the sum of two quartic powers, or generally divide it in this respect. I am sure that I have found a wonderful proof, but unfortunately the space here is too small to write. " 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, Germany Vlfsk announced that it would give 65438+ million marks as a prize to the first person who proved the theorem within 100 years after his death. 1983, gerd faltings proved the model conjunction, and concluded that when N >: 2 (n is an integer), there is no coprime A, B, C, which makes an+bn = cn. 1986, Gerhard Frey put forward "ε conjecture": If A, B and C make an+bn = cn, that is, Fermat's last theorem is wrong, then the elliptic curve y2 = x(x-an)(x+bn) will be a counterexample of Taniyama's conjecture. 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 conjecture in a special case, and the Frey elliptic curve is just within 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. Their proof was published in the Mathematical Yearbook of 1995.
Open classification:
Three major mathematical problems in the modern world: Fermat's last theorem-
Three Mathematical Problems in the Modern World: Fermat's Last Theorem
The New York Times, a recognized world newspaper, published a headline on June 24th, 1993.
About the news that the math problem has been solved, the news headline is "In the ancient math dilemma, someone finally called"
I found it. " The opening article of the first edition of The Times also attached a picture of long hair and wearing a medieval European robe.
Pictures of men. This ancient man was the French mathematician Pierre de Fermat.
Please refer to the appendix for the biography). Fermat is one of the most outstanding mathematicians in17th century, and he has made great achievements in many fields of mathematics.
Great contribution, because he is a professional lawyer, in order to commend his mathematical attainments, the world called him "amateur prince"
"Reputation, one day more than 360 years ago, Fermat was reading a book by the ancient Greek mathematician Diofendus.
When I was writing a math book, I suddenly wrote a seemingly simple theorem in the margin of the page.
Capacity is a problem about the positive integer solution of equation x2+y2 =z2. When n=2, it is called Pythagorean rule.
Li (also called Pythagorean Theorem in ancient China): x2+y2 =z2, where z represents the hypotenuse of a right angle, and X and Y are it.
The square of the hypotenuse of two strands, that is, a right triangle, is equal to the sum of the squares of its two strands. Of course, this equation has
Integer solutions (in fact, there are many), such as: x=3, y=4, z = 5；; x=6、y=8、z = 10； x=5、y= 12、z= 13…
Wait a minute.
Fermat claims that when n>2, there is no integer solution satisfying xn +yn = zn, such as the equation x3 +y3=z3.
Find an integer solution.
Fermat didn't explain the reason at that time, he just left this narrative, saying that he found the proof of this theorem wonderful.
Method, but there is not enough space on the page to write it down. The founder Fermat therefore left an eternal question, 300
Over the years, countless mathematicians have tried in vain to solve this difficult problem. This Fermat, known as the century problem, is the most
The post-theorem has become a big worry in the field of mathematics, and it is eager to solve it quickly.
In the19th century, the Francis Institute of Mathematics in France provided a gold medal and two prizes in 18 15 and 1860.
Whoever solves this difficult problem will be given 300 francs, but unfortunately no one will get a reward. German mathematician Wolff
Skell (p？ Wolfskehl) provides100000 mark in 1908 to those who can prove the correctness of Fermat's last theorem.
The validity period is 100 year. At the same time, due to the Great Depression, this award has depreciated to 7500 marks, although
This still attracts many "math idiots"
After the development of computers in the 20th century, many mathematicians can prove that this theorem holds when n is large.
1983, the computer expert Sloansky ran the computer for 5782 seconds, which proved that Fermat's last theorem was correct when n was 286243- 1.
(Note 286243- 1 is astronomical, with about 25960 digits).
Nevertheless, mathematicians have not found a universal proof. However, this unsolved mathematical problem for more than 300 years has finally been solved.
Yes, this math problem was solved by British mathematician andrew wiles. In fact, Willis is
The development of abstract mathematics in the last 30 years of the 20th century proves this point.
In 1950s, Yutaka Taniyama, a Japanese mathematician, first put forward a conjecture about elliptic curvature, which was later recorded by another mathematician.
Muragoro carried it forward. At that time, no one thought that this conjecture had anything to do with Fermat's last theorem. In the 1980s, Germany
Frey, a mathematician in China, linked Yutaka Taniyama conjecture with Fermat's Last Theorem, and what Willis did was based on this connection.
Prove that one form of Yutaka Taniyama's conjecture is correct, so is Fermat's last theorem. This conclusion
Officially published by Willis1June 2, 9931at the seminar of Newton Institute of Mathematics, Cambridge University, USA. This newspaper
The report immediately shocked the whole mathematics field, and even the public outside the mathematics door wall paid infinite attention. But Willis's
The certificate was immediately found to have some defects, so it took Willis and his students another 14 months to correct it.
Correct it. 1September 1994 19 They finally handed over a complete and flawless scheme, and the nightmare of mathematics finally ended. 1997 6
In May, Willis won the Wolfskeil Prize from the University of G? ttingen. At that time,100000 FAK was about $2 million.
However, when Willis received it, it was only worth about $50,000, but Willis has gone down in history and will be immortal.
Prove Fermat's last theorem is correct
(that is, xn+yn = zn has no positive integer solution to n33)
Just prove that x4+ y4 = z4, xp+ yp = zp (P is an odd prime number) has no integer solution.
Appendix: Biography of Fermat
Pierre de Fermat, one of the greatest mathematicians in17th century, was born in the south of France on August 20th, 160 1.
In a small town near Toulous, his father was a leather merchant, and he died in June 1665+ 10/2.
Fermat university specialized in law, became a professional lawyer after graduation, and served as a member of parliament in Toulouse.
Fermat is a well-read scholar who is proficient in several languages and has a strong knowledge of mathematics and physics.
Interest is a versatile person. Although he began to study mathematics seriously when he was nearly 30 years old, he was not interested in mathematics.
His contribution earned him a reputation as an amateur prince. This title is just enough to commend him.
He made first-class achievements in the field of mathematics, introduced analytic geometry before Descartes, and developed calculus.
There are great contributions at the exhibition, especially Fermat and Pascal are recognized as pioneers of probability theory.
However, what people talk about is his masterpiece in number theory, such as Fermat's Last Theorem (also known as Fermat's Little Theorem,
Different from Fermat's last theorem): apo a(modp) holds for any integer A and prime number P, and this theorem is the first time.
Now in a letter from 1640, the proof of this theorem was later published by Euler. Fermat is very modest,
He was not rich and famous, and he rarely published papers before his death. Most of his works are found in letters with friends and personal letters.
Yes, but usually there is no evidence. The most famous is Fermat's last theorem, and his natural intuition is really different.
Chang Minrui, other theorems he asserted, were later proved by people one after another. The prescient Fermat is actually mathematics.
A wonderful flower in history.
Fermat's last theorem
1On June 24th, 993, The New York Times, a world-recognized authoritative newspaper, published a news about solving mathematical problems. The headline of the news is "In the ancient mathematical dilemma, someone finally shouted' I found it'". The opening article of the first edition of The Times is accompanied by a photo of a man with long hair and wearing a medieval European robe. This ancient man was the French mathematician Pierre de Fermat (please refer to the appendix of Fermat's biography). Fermat is one of the most outstanding mathematicians in17th century. He has made great contributions in many fields of mathematics because he is a professional lawyer. In recognition of his mathematical attainments, the world called him "amateur prince". One day more than 360 years ago, Fermat was reading a math book by Diofendos, an ancient Greek mathematician, when he suddenly had a whim and wrote a seemingly simple theorem. The content of this theorem is about the positive integer solution of an equation x2+y2 =z2. When n=2, it is the well-known Pythagorean Theorem (also called Pythagorean Theorem in ancient China): x2+y2 =z2, where z represents the hypotenuse of a right triangle, and x and y are its two branches, that is, the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two branches. Of course, this equation has an integer solution. x=6、y=8、z = 10； X=5, y= 12, z= 13… and so on. Fermat claims that when n>2, there is no integer solution satisfying xn +yn = zn, such as the equation x3 +y3=z3. At that time, Fermat did not explain why. He just left this narrative, saying that he found a wonderful way to prove this theorem, but there was not enough space on the page to write it down. Fermat, the initiator, thus left an eternal problem. For more than 300 years, countless mathematicians have tried in vain to solve this problem. This Fermat's last theorem, known as the century's difficult problem, has become a big worry in mathematics and is extremely eager to solve it. /kloc-in the 0/9th century, Francis Institute of Mathematics in France provided a gold medal and 300 francs to anyone who solved this problem twice in 18 15 and 1860. Unfortunately, no one can receive the prize. German mathematician Wolfskeil (p? Wolfskehl) provides 100000 marks in 1908 to those who can prove the correctness of Fermat's last theorem, and the validity period is100 years. In the meantime, due to the Great Depression, the bonus amount has been devalued to 7500 marks, but it still attracts many "math idiots". After the development of computers in the 20th century, many mathematicians can prove that this theorem holds when n is large. 1983, the computer expert Sloansky ran the computer for 5782 seconds, which proved that Fermat's last theorem was correct when n was 286243- 1 (Note 286243- 1 is an astronomical figure with about 25960 digits). Nevertheless, mathematicians have not found a universal proof. However, this 300-year-old math unsolved case has finally been solved. Andrew wiles, an English mathematician, solved this mathematical problem. In fact, Willis used the achievements of the development of abstract mathematics in the last 30 years of the twentieth century to prove it. In 1950s, Yutaka Taniyama, a Japanese mathematician, first put forward a conjecture about elliptic curvature, and later another mathematician Goro Shimamura developed this conjecture. 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 Willis 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, Willis' proof was immediately found to have some defects, so it took Willis 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, Willis won the Wolfskeil Prize at the University of G? ttingen. At that time,1100,000 grams was about $2 million, and when Willis received it, it was only worth about $50,000, but Willis has been recorded in the history books and will be immortal. In order to prove that Fermat's last theorem is correct (that is, xn+yn = zn versus n? 3 There is no positive integer solution) Just prove that x4+ y4 = z4 and xp+ yp = zp (P is an odd prime number) have no integer solution. Appendix: Fermat Pierre de Fermat's biography is one of the greatest mathematicians in17th century. He was born in a small town near Tulus in the south of France on August 20th, 160 1. His father was a leather dealer, 16 12. Fermat university specialized in law, became a professional lawyer after graduation, and served as a member of parliament in Toulouse. Fermat is a well-read scholar who is proficient in several languages and has a strong interest in mathematics and physics. He is a man of many talents. Although he began to study mathematics seriously when he was nearly 30 years old, his contribution to mathematics earned him the reputation of an amateur prince. This title is just enough to commend his first-class achievements in mathematics. He introduced analytic geometry before Descartes and made great contributions to the development of calculus. Fermat and Pascal, in particular, are recognized as pioneers of probability theory. However, people are talking about his representative works in number theory, such as Fermat's Last Theorem (also known as Fermat's Little Theorem, which is different from Fermat's Last Theorem): ap? A(modp) holds for any integer A and prime number P. This theorem first appeared in a letter in 1640, and the proof of this theorem was later published by Euler. Fermat was modest, not seeking fame and fortune, and rarely published papers before his death. Most of his works are found in letters and personal notes with friends, but they are usually not confirmed. The most famous is the so-called Fermat's Last Theorem. Fermat's natural intuition is really keen, and other theorems he asserted were later proved by people one after another. Fermat with foresight is really a wonderful flower in the history of mathematics.