On the other hand, the cube of a number cannot be expressed as the sum of two cubes, and a quartic degree cannot be expressed as the sum of two quartic degrees; Or more generally, except for the square, an n-power number cannot be expressed as the sum of two n-power numbers (xn+yn = Zn). I found a wonderful proof for this proposition, but there is not enough space for me to write it down. 」
It was this mysterious announcement that made countless mathematicians in later generations busy providing what Fermat called "wonderful proof". On the surface, when n≥3, xn+yn = Zn has no integer solution. This statement seems simple, but it must not be underestimated. All other theorems mentioned by Fermat were proved or overturned around the beginning of19th century. Just this seemingly simple narrative, no one can answer it, so it was named "Fermat's Last Theorem". Is this theorem true or false? In this century, someone tried to verify this theorem by computer; Basically, the computer can check quite a few numbers, but it still can't check all the numbers. This is the dilemma. Even if this theorem holds for billions of numbers, there are countless numbers and powers behind billions to be verified. Therefore, to declare this theorem valid, we need a mathematical proof. In the19th century, French and German academies of science provided huge bonuses to seek the proof of this theorem. Every year, thousands of professional and amateur mathematicians send strange "proof" methods to mathematical magazines and councils, but the results are all in vain.
.65438+July-August, 0993: Fatal Vulnerability
When wiles walked off the platform on that Wednesday in June, mathematicians were cautiously optimistic. The mystery of 350 years seems to have finally been solved. Many theories and symbols used by wiles were unheard of in Fermat's time, and some even appeared in the 20th century. These theories still need expert certification, so the proof was sent to many top mathematicians. Maybe wiles's seven years of seclusion can finally pay off. But this optimism did not last long. Within a few weeks, wiles's logic was found to be flawed, and he tried to make up for it, but in vain. Peter Sarnak, a mathematician in Princeton, watched in pain as his close friend wiles Zhenri faced the proof he gave to the world in Cambridge two months ago. He explained, "It seems that wiles wants to spread a huge carpet on the floor of the room. When this side is paved, the carpet on the other side of the room will be rolled onto the wall; At the other end, pull the carpet back to the ground and it will arch up somewhere in the room. And whether this blanket is suitable for this room, he can't decide at all. " Wiles went back to his attic. The New York Times and reporters from major media also spared him for the time being. Then, as the days passed, the proof results never appeared, which once again made the mathematics community and the general public begin to doubt whether Fermat's Last Theorem was established. Wiles's beautiful proof of the world, like Fermat's "beautiful proof, but there is no room in the margin", is illusory.
There is such a math problem: although it looks simple, the difficulty is beyond the reach of ordinary people.
Fictitious; This problem is far more famous than mathematics, and almost everyone who is educated knows it.
, although may not understand its specific content; It is far better to make a little progress on this difficult problem than Goldbach.
To make a simple guess, many people can easily make some achievements when they first set foot in it, but if you want to
Completely proved almost impossible.
This mathematical problem was put forward more than 300 years ago, which attracted countless mathematicians to spend their lives trying to find the answer.
Find evidence. The Institute of Mathematics of the former University of G? ttingen in West Germany specially set up a Wall for it at 1908.
Wolfskeil Prize. Although the application papers for this award have many strict regulations, they are quite
For a long time, the graduate school still received an application paper every week on average. This is better than the first prize.
One year has been much better. In that year, 62 1 applications were received! Although in the last decade of this century
Finally, someone gave a complete proof of this mathematical problem, but it took three and a half generations for human beings.
The age of Ji made it a legend in the history of mathematics.
-This mathematical problem is Fermat's Last Theorem. No one can imagine,
The origin of this problem is just a short scribble in the margin of a book!
Fermachi
Pierre de Fermat was born in France in 65438. When he was at 1665
When 65438+ 12 died in 10, he was the most famous mathematician in Europe at that time. But judging from his life, he is not based on
A professional mathematician who lives by mathematics is a lawyer and judge in Toulon, that is, he is responsible for hearing cases.
Officials. When he got this law post at the age of 30, he began to study mathematics in his spare time. although
Fermat had no professional training in mathematics in the past, but he soon showed his talent in mathematics.
The short mathematical experience has added a brilliant page to the whole history of mathematics. Today, his name is often associated with.
However, because most of his work in this field is ahead of the times, his contemporaries are even more.
What I know better is that he is independent of the coordinate geometry invented by Descartes and passed through Newton and Leibniz.
(Leibniz) and others' world-famous calculus, and calculus founded by him and Pascal.
Probability theory. Fermat lived in an era where mathematicians gathered, such as Descartes and Pascal. expense
Ma keeps extensive communication with them and often communicates with them on some math problems. but
Limited to this, Fermat almost never published any mathematical works in his whole mathematical career.
However, this does not hide the dazzling light of these achievements. Take mathematics as a hobby.
With his brilliant achievements in mathematics, he was named "Prince of Amateur Mathematicians".
Around the 3rd century AD, Diophantu, an ancient Greek scholar, wrote Arithmetic, which is one of his most important masterpieces.
. This is the first algebra book written in words. There are some books about integral coefficient equations of two or more variables.
Solving rational numbers is a very important part. Today, mathematicians are generally confined to seeking similar problems.
Its integer solution. In fact, on this issue, there is not much difference between the concepts of rational number and integer. because
For example, one set of solutions of equation 2x+3y = 0 (x = 1/2, y =- 1/3) and another set of its solutions (x = 3).
Y =-2) doesn't make much difference. As long as the first set of solutions is multiplied by the least common multiple whose denominator is 6, then
A second set of solutions can be obtained. So in many cases, our research on this kind of problems is limited to finding integer solutions.
/kloc-in the middle of the 0/5th century, the war made Constantinople fall into the hands of the Turks. In order to seek peace, a large number
Byzantine scholars fled to the west and brought with them many academic works of Greek scholars, including this one.
Arithmetic. However, due to language and other reasons, no one noticed at first. until
162 1 year, Claude Basie published Latin translations, notes and comments.
The new edition of Arithmetic has attracted the attention of European mathematicians. Fermat is one of them.
A scholar with strong interest.
When reading this book, Fermat often used to write some short notes in the margin of the page.
Until the fifth year after his death, his son Samuel was collecting and sorting out his father's pens.
Records and letters were found in preparation for publication. Among them, in the eighth question of Diophantine, "Given a flat
Square number, write it as the sum of the other two squares ",Fermat wrote in Latin:
"On the other hand, it is impossible to write a cubic number as the sum of two cubic numbers, or write a number as four.
The power number is written as the sum of the other two quarts. Generally speaking, for any number, as long as its power refers to it.
If the number is greater than 2, it is impossible to write the sum of two other numbers with the same power exponent. For this proposition, I got
A wonderful proof method, but the space here is too small for me to write them down. "
Mathematically, the eighth problem of Diophantine is that X2+Y2 = Z2 has a positive integer solution.
It has been said that this kind of problem only needs positive integer solutions.
Fermat thinks that equations x3+y3 = z3 and x4+y4 = z4 have no positive integer solutions. here
On this basis, Fermat draws the conclusion that the equation xn+yn = Zn (n ≥ 3) does not have a positive integer solution.
Therefore, Fermat's last theorem seems to have a little mystery. Unlike Goldbach's conjecture
For example, Fermat's Last Theorem has been called "Theorem" since its appearance, although it has been more than 300 now.
No one has been able to get Fermat's idea for years, but it can't be recorded just because the blank is too small.
Proof, some people have always doubted whether Fermat himself really got the proof of this proposition, but he never did.
No one has ever doubted the correctness of this theorem. Why is this theorem called "the last theorem"? also
There is no textual research. From some known information, it can be concluded that this annotation should be Fermat in the 30 th century in 17.
Written one day in the s, this is definitely not the last mathematical conclusion in Fermat's mathematical career.
. So more people think that this "last theorem" is named because it was left by Fermat.
The last mathematical theorem to prove!
Starting with the Pythagorean number
There is a mathematical theorem that China people are very familiar with, which is called Pythagorean Theorem and a simple solution to "Hook three strands, four chords and five".
Interpretation is what many children blurt out when they are learning mathematics. In fact, Diophantine's eighth question says
"Given a square number, write it as the sum of the other two squares" is the opposite line, such as x2+y2 = z.
Study on solving the equation of 2. A set of solutions (x, y, z) of this equation is a set of pythagorean numbers. identical
The theorem is called Pythagoras theorem in the west, and Pythagoras number is Pythagoras number.
. (Because Fermat's Last Theorem was put forward by western mathematical circles, we used western names when we studied here.
Oh! ) Once we get a set of Pythagorean numbers, we can get countless other sets of Pythagorean numbers.
Number, you just need to multiply the solutions of this group of different coefficients. For example, 2 times 3, 4, 5.
6,8, 10 is also a set of Pythagorean numbers. Because 62+82 = 102 is relatively simple, we changed it from 32+42 = to.
52 It can be deduced that 32× m2+42× m2 = 52× m2, that is, (3m) 2+(4m) 2 = (5m) 2.
A complete answer has been found in the book "The Elements" written by Euclid from about 350 to 300 BC.
The content of the norm diagram problem. Let x = S2-T2, y = 2st and z = S2+T2, where s,
T is an arbitrary natural number as long as s is greater than t and they have no common factor.
For most people, this theorem is hardly difficult. Let's try to take a step or two.
Look! When n = 4, does the equation: x4+y4 = Z4 have a solution? In the proof of a mathematical theorem
In this process, people usually try to draw some conclusions with some special situations first, and then get completeness.
The answer. What we have done is just such an attempt. In the proof of mathematical propositions, we all know that there is one
This method is called reduction to absurdity, that is, starting from the opposite side of the proposition, first assume a conclusion opposite to the proposition, and then
Infer contradictions from assumptions. Once it is proved that the negative proposition of a proposition is not established, the original proposition can be obtained.
The conclusion of establishment. For this reason, we assume that the equation x4+y4 = Z4 has a solution when n = 4. According to this set of solutions
We can take a = Y4, b = 2x22, c = Z4+X4 and d = Y2XZ.
Next, we repeatedly use the well-known identity (R+S) 2 = R2+2RS+T2 to get it.
：
a2+b2=(z4-x4)+4x4z4
=z8-2x4z4+x8+4x4z4
=(z4+x4)2
=c2
We have:
( 1/2)ab =( 1/2)y42x2z 2 =(y2xz)2 = D2( 1)
What we want to prove now is that the formula (1) is wrong. Here, we will use another method.
It was also created by Fermat himself, and it is called infinite descent method. As we all know, using a set of Pythagorean ternary numbers
For the three sides of a triangle, you can get a right triangle, which is called a pythagorean triangle for short. Fermat
It is proved that the area of Pythagorean triangle can never be a square number, that is, it will never be the square of an integer. Proved as follows:
Suppose there is a pythagorean triangle whose area is exactly the square of the integer u, and the other x, y and z groups.
Pythagorean number is the length of three sides of a triangle, where z is the hypotenuse. From Pythagorean Theorem, we can get: x2+y2 = z2.
Then, from the right triangle area formula, we can get
u2=( 1/2)xy (2)
Note that formula (2) here is essentially equivalent to formula (1). Fermat's other clever argument makes us have to
Know that there must be another set of solutions x, y, z, u, so: X2+Y2 = Z2, U2 = (1/2).
XY and z > z
At this point, the contradiction we need is at our fingertips. In the same way, we can always get countless
Xn, Yn, Zn and UN (n = 1, 2, 3 ...), and there is z > z > z1> z.
2 > z3 > ... an array of positive integers that can descend indefinitely. But in fact, there is no infinite decline.
Of an array of positive integers. Because when Zn drops to 1, it can't drop any more!
Therefore, we come to the conclusion that formula (2) is not valid. In other words, the formula (1) is also invalid. In this way,
When n = 4, we get the proof of Fermat's last theorem. A simple inference enables us to continue.
Take a small step, that is, for all n = 4k, Fermat's Last Theorem holds. The reason is that if equation X4
K+Y4K = Z4K has solutions A, B and C, then ak, bk and ck will be the equation X4+Y4 =
A set of solutions of Z4. And we have proved that it is unsolvable. In this way, we can easily stand in Fermat's position.
Part of the proof of Fermat's Last Theorem under special circumstances is obtained on the shoulder.
Hard exploration
Looking back on the last section, you may ask, why don't we try the case of n = 3? However, when you taste
Give it a try and you will understand why. It is more difficult to prove Fermat's Last Theorem when n = 3 than when n =
The situation at 4 o'clock.
1753 On August 4th, Euler wrote to Goldbach. In the letter, he announced the evidence of success.
Fermat's last theorem when n = 3 is given, but it is not proved. 17 years later, when Euler was in St. Petersburg,
Only when he published Introduction to Algebra did he prove that it still had serious defects. Fortunately, yes.
For n = 3, this defect is not irreparable. But if you try to continue to give it in Euler's way,
Other special values prove that this error is fatal.
Euler also used the infinite descent method. He constructed a line for this, such as:
Where a and b are integers. Then Euler found the spear he needed through a series of transformations.
Shield, and introduced the conclusion that the original proposition was established. Although there is nothing wrong with this algebraic transformation process, but
He cursed it when he first built this array. Euler's numbers take different values for a and b.
Form a digital system. In the proof, Euler naturally applied some characteristics of integer system to the new
In the digital system, but in fact this analogy is not valid. Although for some special values in two series
N = 3 is one of them, it does have the same properties, but it can't get a general knot.
Open. So Euler relied more on luck when giving this proof. If he wants n = 5.
It is proved that according to his method, more complex numbers need to be constructed. And at this time, Euler himself will definitely
Be aware of your mistake.
Now we have obtained the proof of Fermat's Last Theorem about another special value. Let's sum up one.
Go down. Just like proving the inference of N = 4, we also have: X3K+Y3K = Z3K without solution.
. On the basis of these two further inferences, we can simplify the proposition of Fermat's Last Theorem.
Melt it. We consider "fundamental theorem of arithmetic": every natural number greater than 1 is either a prime number or a prime number.
It is expressed as the product of several prime numbers, which is unique if the order of prime numbers is not included. because
N in the proposition is ≥ 3, so N can be divisible by both prime numbers greater than 2 and 4 (both are divisible at the same time).
Separability can fall into any of these categories). In this way, the problem is simplified to solving all odd prime numbers.
Proof of (only 2 in prime numbers is even) and n = 4. And n = 4 is the simplest, so what we have to do.
Yes, it is to verify all odd prime numbers.
1825, two mathematicians, one old and one young, proved the final theorem of n = 5. They are 70 years old.
Legendre and 20-year-old Dirichlet. They extended Euler's method.
After giving many assumptions carefully, it proved to be successful. But when n = 5
All the well-known methods are exhausted. It is proved that the demand for algebraic tools is increasing.
Harsh. Dirichlet tried to solve the case of n = 7 but failed. He only got one in 1832.
A rather weak conclusion is that Fermat's Last Theorem holds for n = 14. 1839, Lamy finally proved.
The case of n = 7 is discussed. At this time, in his proof, people must turn to something that is very integrated with 7 itself.
A small and exquisite mathematical tool. He further proved Fermat's last theorem, but at the same time
At that time, all the paths found by human beings on the road to solving this problem were blocked. If you don't accept the new one,
It is hopeless to prove that n = 1 1. 1847, the lame man found another one himself.
A circuitous way forward.
The core of Lame's suggestion is to try to solve Fermat's last theorem once and for all with N complex unit roots. place
The unit root of complex number n refers to a complex number r, which satisfies rn = 1, but for any positive integer less than n,
K, use rk≠ 1 What is the purpose of introducing R? All Fermat's theorems so far
In several examples of proof, without exception, some factorization in algebra is used. If n = 3, it is used.
Factorization formula: x3+y3 = (x+y) (x2-xy+y2)
Lame realizes that the difficulty of proof increases with the increase of n, because when this factorization is carried out, it is
The frequency of one of the decomposition factors is getting higher and higher. Once r is introduced, it is possible to completely transfer xn.
+yn is decomposed into n factors, all of which are 1 time.
On March 1847 and 1 day, Lame, who was extremely excited, made a report to the members of the Paris Academy of Sciences and announced that he had finished it.
Fermat's last theorem has been completely proved. He used the number formed by the concept r he introduced-it is now called R.
It is a cyclotomic integer and an infinite descent method given by Fermat himself. When n = 3, the whole proof is the same as Euler's proof.
The arguments are very similar. After talking about the certificate he found, Lame suggested to him and urged him to finally complete it.
Certificate colleague Lionel expressed his gratitude. However, as he sat down, Lionel pointed out,
The proof of Lame depends on the unique factorization theorem. As far as he knows, this does not exist for cyclotomic integers.
Theorem of.
Leo Orville's speech hit the nail on the head and pointed out the key points of Lame's argument. As if it were a joke,
After Lame, who was sad and embarrassed, spent weeks trying to make up for the failure, Lame realized that,
He made the same hopeless mistake as Euler.
Every dark cloud has a silver lining. The complete destruction of Lame proves that the theory is actually
Kummer, another mathematician, published an article in an unknown journal three years ago.
Documents. If Lame had known the result, he might not have made a mistake. When mela admitted
When he realized his mistakes and Cuomo's achievements, Cuomo had established a brand-new mathematical theory.
Theory, and use it to prove Fermat's last theorem. Also at 1847, Cuomo got an A.
A milestone conclusion: For all prime indexes less than 37 (of course, it is also true for all indexes less than 37)
), and except 37, 59 and 67, Fermat's Last Theorem holds for all prime indices less than 100.
After an extremely difficult journey, people have been helped by computers in this century.
The solution of Fermat's last theorem is very fast. First put forward by Stafford and Vandive.
R) Check all prime numbers less than 6 17. In 1954, Lehmer further checked to 400.
1, and then the number reached 30000. 1976, wagstaff of the United States proved himself right.
Fermat's last theorem applies to all power exponents less than 125000.
1983 At the beginning of this year, 29-year-old German mathematician gerd faltings proved a conclusion.
It marks that the most famous unsolved problem in mathematics has made the greatest progress in 100 years. It turns out that he is right.
For every exponent n greater than 2, the Fermat equation has at most a finite number of primitive solutions (that is, solutions without common factors).
This certificate helped faltings win the Fields Prize of 1986, but people didn't know it.
Ming Can leads to the complete proof of the last theorem? But in any case, faltings has infinite existence.
The possibility of a solution is reduced to a limited number of solutions at most, which is indeed a qualitative leap!
The Final Proof of Theorem
Although in the eyes of ordinary people, everyone believes that Fermat did find a proof. But it looks more like
A touching story. /kloc-an amateur mathematician in the 0/7th century, formed a proof of a proposition in his mind.
It has made countless professional mathematicians struggle for it for more than three centuries. Fortunately, in humans,
In the last decade of the next century, the attractive veil of Fermat's last theorem has finally been unveiled!
The final attack route is completely different from Fermat himself, Euler and Cuomo, and it is modern mathematics.
Many branches (such as elliptic curve theory, model theory, Galois representation theory, etc. ) are all integrated.
The result of the action. Because the whole proof process involves many profound mathematical theories, many mathematicians do so.
Made a contribution. We can't go into details here, we can only outline the proof route very roughly.
In the fifties and sixties of this century, an important conjecture gradually formed in the study of number theory, which was first proposed by.
Yutaka Taniyama (Y. Taniyama) proposed that following Goro Shimomura and A. Weil (we
Il) is refined into the following form: Every elliptic curve on the rational number field is a modular curve. (Now it is generally called.
It is called the Gushan-Zhicun conjecture. )
Since the late 1960s, some people have compared the Fermat equation Xn+Yn = Zn with the form Y2 = X (X+.
A) (x+b), the initial emphasis is on the application of Fermat's last theorem.
Conclusion Prove the conclusion related to elliptic curve. From 65438 to 0985, Frey took a step forward in the connection between the two.
Taking an important step, he proposed that if Fermat's last theorem was not established, it would conflict with the Gushan-Zhicun conjecture.
Shield. 1986, other mathematicians continued to demonstrate on this basis, and finally put Fermat's last theorem.
This proof comes down to the proof of intellectual village conjecture of Taniyama.
1June, 993, the British mathematician wiles experienced seven years of struggle.
Later, at the mathematics seminar held by Newton Institute of Mathematics at Cambridge University, he announced that he had proved Taniyama-Zhicun.
Guess, on this basis, wiles announced that he had proved Fermat's Last Theorem. However, history is always amazing.
The Tao is in samsara. While demonstrating wiles's proof of more than 200 pages, mathematicians found that.
There is a loophole! 199365438+On February 4th, wiles sent an email to his colleague, confirming his certificate.
There were mistakes in the Ming dynasty. Does this mean that scientists who are about to enter the 2 1 century must submit to it more than three centuries ago?
At the feet of an amateur mathematician?
The answer is no, once again, human beings have achieved the goal of constantly surpassing themselves with actions. What crime did wiles commit?
Under the error, by himself to give supplementary proof. 1994101October 25th, Ohio State University, USA.
Professor Rubin announced cautiously and optimistically to his friends in the mathematics field by e-mail: "wiles is finished."
Proved Fermat's last theorem! "
In the July issue of 1995, the Bulletin of the American Mathematical Society published an article entitled "Taylor".
And wiles's proof of Fermat's last theorem ". At the beginning of the article, the author declared in an extremely positive tone: "In
The conjecture mentioned in this paper was finally completely proved in September 1994! "At this point, people can be sure.
I believe that the famous "theorem" that has puzzled mathematicians for more than 300 years has really become a theorem!
The story of Fermat's Last Theorem is unique in the history of science. From the day it was put forward, it was
The title "Theorem" is doomed to be different. And people seek its complete solution, it seems that only.
Because I don't trust Fermat. But that's it? Don't! Mathematicians finally pursued Fermat.
The proof of the theorem once again shows that the attitude towards science must be rigorous and there is no ambiguity allowed. because
The structure of our human society building can only be built on a solid scientific basis!