Euler’s Conjecture (also known as Fermat’s Last Theorem)
Content
When the integer n > 2, about x, y, z The indeterminate equation x^n + y^n = z^n has no positive integer solutions.
Or when ∀b>2, n-1∑aⁿi has no positive integer solution.
Introduction
This theorem, originally called Fermat’s Last Theorem, was put forward by the French mathematician Fermat in the 17th century, and it was called "Theorem" at that time. , Not really believe that Fermat has proved it. Although Fermat claimed that he had found a wonderful proof, Fulfske of Germany announced a bonus of 100,000 marks to be the first person to prove the theorem within one hundred years after his death. This attracted many people to try and Submit their "certificate". After the First World War, Mark depreciated sharply, and the charm of the theorem also greatly declined.
But after three and a half centuries of hard work, this problem of number theory in the century was successfully proved by Princeton University British mathematician Andrew Wiles and his student Richard Taylor in 1994. The proof uses a lot of new mathematics, including elliptic curves and modular forms in algebraic geometry, as well as Galois theory and Hecke algebra. It is doubtful whether Fermat has really found the correct proof. Andrew Wiles won the 1998 Fields Medal Special Award and the 2005 Shaw Prize in Mathematics for successfully proving this theorem.
Origin
In 1621, when 20-year-old Fermat was reading a Latin translation of "Arithmetic" by the famous Greek mathematician Diophantus in the third century AD, he was on the 11th. Next to the proposition of Volume 8, there is a paragraph on the page about all the positive integer solutions of the indefinite equation x2+y2=z2 , In a nutshell: "An equation of the form xn+yn=zn. When n>2, there can be no integer Solution. Regarding this, I am sure that I have discovered a wonderful proof method, but unfortunately the blank space here is too small to write." (Latin original: "Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.")
After all, Fermat did not write down the proof, and his other conjectures contributed a lot to mathematics, which inspired many mathematicians to be interested in this conjecture. The related work of mathematicians has enriched the content of number theory and promoted the development of number theory. For many different n, Fermat's theorem has long been proved. However, no one has obtained a universal method of proof. Over the past three hundred years, countless scholars have paid a lot of effort to prove this conjecture, but they can neither prove nor deny it.
Research history
n=3
Euler proved that n=3, using the unique factorization theorem.
n=4
Fermat himself proved that n=4.
n=5
In 1825, Dirichlet and Legendre proved that n=5, using an extension of Euler’s method, but avoiding the only Factorization theorem.
n=7
In 1839, French mathematician Lame proved the case of n=7. His proof used a clever tool closely integrated with 7 itself, but It is difficult to generalize to the case of n=11; therefore, in 1847, he proposed the "circular integer" method to prove it, but it was unsuccessful.
Ideal number
Kummer proposed the concept of "ideal number" in 1844. He proved that Fermat's last theorem holds for all prime indices n less than 100. One research has come to an end.
In 1849, the German mathematician Kummer introduced the concept of "ideal number" he discovered using the method of modern algebra, and pointed out that Fermat's problem is only possible when n is equal to a certain value. Incorrect, so only these values need to be studied. Although he spent his life studying this problem, although he did not finally solve it, he proposed a set of mathematical theories, which promoted the development of mathematics.
By the first half of the 20th century, mathematicians pushed the proof to an odd number n=619. In 1976, American mathematicians proved that the odd number of 2125000 also proved quite a lot. It is said that the largest odd prime number n is close to Around 41 million.
Mödel’s Conjecture
In 1922, the British mathematician Model put forward a famous conjecture, which is called the Model Conjecture. According to its original form, this conjecture says that any irreducible, rational coefficient of a binary polynomial, when its "genus" is greater than or equal to 2, there are at most finite solutions. Let this polynomial be f(x, y), and the conjecture shows that there are at most finite logarithmic even xi, yi ∈ Q, so that f(xi, yi)=0.
Later, people extended their conjecture to polynomials defined in arbitrary number fields, and with the emergence of abstract algebraic geometry, they re-used algebraic curves to describe this conjecture. Therefore, what Faltings actually proved is: arbitrarily defined on the number field K, the algebraic curve with a genus greater than or equal to 2 has at most a finite number of K points.
Mathematicians have given various comments on this conjecture, which generally seem to be negative. In 1979, Liebenbom said: “There are good reasons to believe that the certification of the Model’s conjecture seems to be a distant matter.”
For the “conjecture”, Will criticized in 1980: “Mathematicians often He said to himself: If something is true,'this is great' (or'this is so smooth'). Sometimes it doesn't take much to verify his conjecture, sometimes it quickly denies it. However, if he fails to confirm his prediction after a period of hard work, then he will talk about the word'guess', even if this thing is of no importance to him. In most cases, it is not passed. Well thought out." Therefore, for the Model's conjecture, he pointed out: Let's take a look at the "Mauder's conjecture." It involves a question that arithmetic seldom does not ask; therefore, people do not get any serious revelation as to whether to bet on the right or wrong.
However, not long after, in the summer of 1983, the 29-year-old mathematics lecturer at Wappertal University in Germany, the Federal German mathematician Gerd·Faltings proved that Mo Del’s conjecture has opened a new chapter in the study of Fermat's Last Theorem, and people have a new perspective on it. In Faltings' article, two other important conjectures were also solved at the same time, namely, the Tait and Shavalwicz conjectures, which have the same significance as the Model conjecture. Faltings won the 1986 Fields Award.
Here I mainly explain the Mordel’s conjecture, as for the proof, I won’t talk about it much. The so-called algebraic curve, roughly speaking, is the set of all solutions with f(x, y)=0 in any domain containing K.
Let F(x,y,z) be a homogeneous polynomial of degree d, where d is the degree of f(x,y), and let F(x,y,1)=f(x, y), then the genus g of f(x,y) is
g≥(d-1)(d-2)/2
when f(x,y) Take the equal sign when there is no singularity.
The Fermat polynomial x^n+y^n-1 has no singularity, and its genus is (n-1)(n-2)/2. When n≥4, the Fermat polynomial satisfies the conjecture condition. Therefore, xn+yn=zn has at most a finite number of integer solutions.
Why did the conjecture remove the case where the genus of f(x, y) is 0 or 1, that is, remove the case where the number of times d of f(x, y) is less than or equal to 3? We explain the reason for it.
When d=1, f(x,y)=ax+by+c obviously has infinitely many solutions.
When d=2, f(x,y) may not have a solution, for example, f(x,y)=x2+y2+1; but if it has one solution, there must be infinitely many solutions . Let us prove this point geometrically. Let P be a point in the solution set of f(x, y), and let l denote a straight line that does not pass through the point P (see the figure above). For a point Q with coordinates in domain K on l, the straight line PQ always intersects the solution set at another point R. When Q traverses an infinite number of K-points on l, the set of points R is the infinite set of K-solutions of f(x, y). For example, applying this method to x2+y2-1 gives a well-known parameterized solution:
When F(X, Y, Z) is a cubic non-singular (that is, no singularity) curve, The solution set is a so-called elliptic curve. We can use geometric methods to make an infinite set of solutions. However, for non-singular curves F whose degree is greater than or equal to 4, this geometric method does not exist. Even so, there are high-dimensional algebraic clusters called abelian clusters. Studying these abelian clusters forms the core of Faltings's proof.
Faltings used a large amount of algebraic geometry knowledge such as Shavalvitch’s conjecture, Jacobian clusters, height, homology, and Tait conjecture when proving Model’s conjecture. Moder's conjecture has a wide range of applications. For example, before Faltings, people didn’t know that for any non-zero integer a, the equation y2=x5+a had only a finite number in Q
a finite set of relatively prime
In 1983, Gerd Faltings proved Mordell's conjecture, and concluded that when n > 2 (n is an integer), there is only a finite set of relatively prime a, b, and c such that an + b n = cn.
Gerhard Frey
In 1986, Gerhard Frey proposed the "ε-conjecture": if there are a, b, c such that a^n + b^n = c^n, that is If Fermat's Last Theorem is wrong, then the elliptic curve y^2 = x(x-a^n)(x + b^n) would be a counterexample to the Taniyama-Shimura conjecture. Frey's conjecture was immediately confirmed by Kenneth Ribet. This conjecture shows the close relationship between Fermat's Last Theorem and the elliptic curve and modular form.
Wiles and Taylor
In 1995, Wiles and Taylor proved the Taniyama-Shimura conjecture within a special case, and Frey’s elliptic curve happened to be within the scope of this special case. , Which proves Fermat's last theorem.
Wiles
Wiles' proof of Fermat's Last Theorem is also very dramatic. It took him seven years to derive most of the proof under unknown circumstances; then he announced his proof in a seminar at Cambridge University in June 1993, and it immediately became the headline of the world. However, in the process of approving the certificate, the expert discovered a flaw. Wiles and Taylor then improved it for nearly a year, and succeeded in September 1994 with a method previously abandoned by Wiles. This part of the proof is related to Iwasawa's theory. Their proof was published in the 1995 Annals of Mathematics (en:Annals of Mathematics).
Taniyama-Shimura Conjecture
In 1955, Japanese mathematician Taniyama Toyoda first guessed that elliptic curves existed between another type of mathematicians who knew more about curves-modular curves There is a certain connection; Taniyama’s conjecture was further refined by Wei Yi and Shimura Goro to form the so-called "Taniyama-Shimura Conjecture". This conjecture shows that the elliptic curves in the field of rational numbers are all modular curves. This very abstract conjecture has puzzled some scholars, but it has made the proof of "Fermat's Last Theorem" a step forward.
The relationship between the two
In 1985, the German mathematician Frey pointed out the relationship between the Taniyama-Shimura conjecture and Fermat’s Last Theorem; he put forward a proposition: hypothesis "Fermat's Last Theorem" does not hold, that is, there is a set of non-zero integers A, B, C, such that A's nth power + B's nth power = C's nth power (n>2), then use this set of numbers The constructed elliptic curve shaped like the square of y=x(x+A to the nth power) multiplied by (xB to the nth power) cannot be a modular curve. Although he worked hard, his proposition and "Taniyama" ——Shimura Conjecture" contradiction, if these two propositions can be proved at the same time, according to the contradiction method, it can be known that "Fermat's Last Theorem" is not true, this assumption is wrong, and thus the "Fermat's Last Theorem" is proved. But at that time he There is no strict proof of his proposition.
Frey's Proposition
In 1986, American mathematician Bethe proved Frey's proposition, so he hoped to focus on "Taniyama-Shimura Conjecture" .
Complete the proof
In June 1993, the British mathematician Wiles proved that for a large class of elliptic curves on the field of rational numbers, the "Taniyama-Shimura Conjecture" was established. Because he showed in the report that the Frey curve happened to belong to the big type of elliptic curve he said, it also showed that he finally proved the "Fermat's Last Theorem"; but the expert's examination of his proof found that there were loopholes, so After more than a year of hard work, Wiles successfully proved the "Fermat's Last Theorem" in September 1994.
Four-color Theorem
Content and Proposal< /p>
The content of the four-color problem is: "Any flat map only uses four colors to make countries with common borders colored in different colors. "It is expressed in mathematical language, that is, "The plane is arbitrarily subdivided into non-overlapping areas. Each area can always be marked with one of the four numbers 1, 2, 3, and 4, without causing adjacent Both areas get the same number. "
The adjacent area referred to here means that a whole section of the boundary is common. If two areas only meet at one point or a limited number of points, they are not called adjacent. Because the same Coloring them will not cause confusion.
The four-color conjecture came from the United Kingdom. In 1852, when Fernancis Guthrie, who graduated from the University of London, came to a scientific research unit to do map coloring work, An interesting phenomenon was discovered: "It seems that each map can be colored with four colors, so that countries with common borders are colored in different colors. "Can this phenomenon be rigorously proved mathematically? He and his brother Gris, who was studying at the university, are determined to give it a try. The two brothers have used a pile of manuscripts to prove this problem, but the research work has not progressed. .
The process of verification
On October 23, 1852, his brother consulted his teacher, the famous mathematician De Morgan on the proof of this question, and Morgan did not have it. I was able to find a way to solve this problem, so I wrote to his friend, the famous mathematician Sir Hamilton. After receiving Morgan’s letter, Hamilton demonstrated the four-color problem. But until Hamilton's death in 1865, there was no problem. It can be solved.
In 1872, Kelly, the most famous mathematician in the UK at the time, formally raised this question to the London Mathematical Society, so the four-color conjecture became a problem of concern to the world's mathematics community. Many first-class mathematics in the world Everyone has participated in the battle of the four-color conjecture. Between 1878 and 1880, the famous lawyers and mathematicians Kemp and Taylor submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. Everyone I think the four-color conjecture will be solved from then on.
Kemp’s proof is this: first point out that if no country surrounds other countries, or if no more than three countries meet at one point, this kind of map will be It is said to be "regular" (left picture). If it is a regular map, otherwise it is an informal map (right picture). A map is often linked by a regular map and an informal map, but the color required for an informal map The number generally does not exceed the colors required by the regular map. If there is a map that requires five colors, it means that its regular map is five-color. To prove the four-color conjecture, it is enough to prove that there is no regular five-color map.
Kemp uses reduction absurdity to prove it, to the effect that if there is a regular five-color map, there will be a "very small regular five-color map" with the least number of countries. In a regular five-color map, if a country has fewer than six neighbors, there will be a regular map with a small number of countries that is still five-color. In this way, there will be no very small number of countries on the five-color map. There is no formal five-color map. In this way, Kemp thinks that he has proved the "four-color problem", but then people find him wrong.
However, Kemp’s proof clarifies two important concepts. It provides a way to solve future problems. The first concept is "configuration." He proved that in every regular map at least one country has two, three, four or five neighbors, and there is no every Each country has a regular map of six or more neighboring countries, that is, a set of "configurations" consisting of two neighboring countries, three neighboring countries, four or five neighboring countries is inevitable , Each map contains at least one of these four configurations.
Another concept proposed by Kemp is "reducibility". The use of the word "reducible" comes from Kemp's argument He proved that as long as a country has four neighbors in a five-color map, there will be a five-color map with a reduced number of countries. After entering the concept of "configuration" and "reducible", some standard methods for checking configuration to determine whether it is reducible have been gradually developed. The ability to find the inevitable group of reducible configurations is an important basis for proving the "four-color problem" . But to prove that a large configuration is reducible, a lot of details need to be checked, which is quite complicated.
Eleven years later, in 1890, Hewood, who was only 29 years old at Oxford University, pointed out the loopholes in Kemp’s proof with his own precise calculations. He pointed out that Kemp's reason for saying that there is no tiny five-color map can have a country with five neighbors is flawed. Soon, Taylor's proof was also denied. People found that they actually proved a weaker proposition—the five-color theorem. In other words, to color the map, five colors are enough. Later, even though more and more mathematicians racked their brains, they found nothing. As a result, people began to realize that this seemingly easy problem is actually a difficult problem comparable to Fermat's conjecture.
Since the beginning of the 20th century, scientists have basically proceeded to prove the four-color conjecture in accordance with Kemp's ideas. In 1913, the famous American mathematician Burkhoff of Harvard University used Kemp's ideas and combined his own new ideas; he proved that certain large configurations are achievable. Later, the American mathematician Franklin proved in 1939 that all maps under 22 countries can be colored in four colors. In 1950, someone advanced from 22 countries to 35 countries. In 1960, it was proved that the maps of 39 countries or less could be colored with only four colors; then it was advanced to 50 countries. It seems that this progress is still very slow.
Success in the information age
The invention of high-speed digital computers has prompted more mathematicians to study the "four-color problem". Heike, who has been studying the four-color conjecture since 1936, publicly declared that the four-color conjecture can be proved by finding the inevitable group of reducible figures. His student Diolei wrote a calculation program. Heike can not only use the data generated by this program to prove that the configuration is reducible, but the method of describing the reducible configuration is to start from transforming the map into a mathematically called "dual" shape. .
He marked the capitals of each country, and then connected the capitals of neighboring countries by a railway that crossed the border, except for the capital (called the vertex) and the railway (called the arc or edge) , Erase all other lines, and the remaining is called the dual graph of the original graph. In the late 1960s, Heike introduced a method similar to moving charges in an electrical network to find the inevitable group of configurations. In Heike's research, the "discharge method" appeared in a rather immature form for the first time. This is the key to future research on the inevitable group and is also the central element to prove the four-color theorem.
After the advent of electronic computers, due to the rapid increase in calculation speed and the emergence of human-computer dialogue, the process of proving the four-color conjecture has been greatly accelerated. Harken at the University of Illinois in the United States set out to improve the "discharge process" in 1970, and later cooperated with Appel to develop a good program. In June 1976, they spent 1,200 hours on two different electronic computers at the University of Illinois, making 10 billion judgments, and finally completed the proof of the four-color theorem, which made a sensation in the world.
This is a major event that has attracted many mathematicians and mathematics enthusiasts for more than 100 years. A special postmark of "four colors are enough" was stamped to celebrate the solution of this problem.
The "four-color problem" proved to not only solve a problem that lasted more than 100 years, but also became the starting point for a series of new thinking in the history of mathematics. In the research process of the "four-color problem", many new mathematical theories have emerged, and many mathematical calculation skills have been developed. For example, turning the coloring of the map into a problem of graph theory enriches the content of graph theory. Not only that, the "four-color problem" has played a role in effectively designing aviation flight schedules and designing computer coding programs.
However, many scientists are not satisfied with the achievements of computers. They believe that there should be a simple and concise method of written proof. Many scholars and mathematics enthusiasts are still looking for more concise proof methods.
Geometric Proof
In a flat map, in order to distinguish adjacent graphics, adjacent graphics need to be colored with different colors, which are adjacent to these two adjacent graphics. The edge graphics need to use the third color. We first assume that the four-color theorem holds. According to the four-color theorem, there are at most four graphics with adjacent edges in a plane, and because the fourth is the same as 三< /b>A graph with adjacent sides has adjacent sides. A graph with adjacent sides surrounds a graph. Therefore, there are at most four graphs with adjacent sides in a plane. Therefore, the four-color theorem holds (with adjacent sides). Edges, for example: Three graphics with adjacent edges—A and B have adjacent edges, C and AB have adjacent edges)
Mathematical proof
Find the proof using mathematical theory It is the ultimate goal of human research on the "four-color problem".
There is an example of theoretical proof of the four-color theorem, and the proof is proved by the second mathematical induction. The general idea is: firstly, verify that the four-color theorem holds when the initial value 1≤n≤15; secondly, set the inductive hypothesis 15≤n≤k when the four-color theorem holds; again, recursively n=k+1 when the four-color theorem holds. Recursive season Q is the state of configuration, which is divided into four types of arguments: two-configuration, three-configuration, four-configuration, and five-configuration.
The theoretical basis of the proof is that Kemp proved that “in every regular map there is at least one country with two, three, four, or five neighboring countries. There are six or more regular maps of neighboring countries." On the basis of this, the concepts of n configuration (n takes 2, 3, 4, 5), configuration country, regular map boundary, border country, etc. are proposed and clarified. The three lemmas about the five configurations are proved by the construction method, the contradiction method, and the second mathematical induction method in turn. Lemma 1: The set of the number of countries in the five configurations W={12,14,15,...,n ,...}; Lemma 2: In any five configurations, the country of configuration is not a border country; Lemma 3: In the five configurations with n≥15, if each neighboring country of Q is surrounded by the country of Q, there is only one For the common boundary, the number of neighboring countries of Q is five. If there is a country P with more than five neighbors among the five neighboring countries, then the four-color theorem holds.
This proof uses "block" color change, which is different from Kemp's "proof" that uses "Kemp chain" color change.
Goldbach’s Conjecture
Conjecture
Of the historical mathematical conjectures related to prime numbers, the most famous is of course the "Goldbach Conjecture".
On June 7, 1742, the German mathematician Goldbach put forward a bold conjecture in a letter to the famous mathematician Euler:
Odd numbers less than 3 can be the sum of three prime numbers (for example: 7=2+2+3, at that time 1 is still a prime number).
In the same year, on June 30, Euler put forward another version of Goldbach’s conjecture in his reply:
Any even number can be the sum of two prime numbers (such as : 4=2+2. At that time, 1 was still a prime number).
This is the famous "Goldbach Conjecture" in the history of mathematics. Obviously, the former is a corollary of the latter. Therefore, just prove the latter to prove the former. Therefore, the former is called the weak Goldbach conjecture (which has been proved), and the latter is called the strong Goldbach conjecture. Since 1 is no longer classified as a prime number, these two conjectures have become
any odd number not less than 7 can be written as the sum of three prime numbers;
Even numbers less than 4 can be written as the sum of two prime numbers.
Introduction
In his reply to Goldbach, Euler made it clear that he was convinced that these two conjectures were correct theorems, but Euler could not give a proof at the time. . Since Euler was the greatest mathematician in Europe at the time, his confidence in Goldbach’s conjecture affected the entire European and world mathematics community. Since then, many mathematicians have been eager to try, and even devoted their life to proving Goldbach’s conjecture. But until the end of the 19th century, the proof of Goldbach’s conjecture did not progress. The difficulty of proving Goldbach’s conjecture is far beyond people’s imagination. Some mathematicians compare Goldbach’s conjecture to "the jewel in the crown of mathematics."
We start with 6=3+3, 8=3+5, 10=5+5,..., 100=3+97=11+89=17+83,...These specific examples It can be seen that Goldbach’s conjecture is all valid. Someone even verified all the even numbers within 33 million one by one, and none of them did not meet Goldbach's conjecture. In the 20th century, with the development of computer technology, mathematicians discovered that Goldbach's conjecture still holds true for larger numbers. But natural numbers are infinite. Who knows if a counterexample to Goldbach’s conjecture suddenly appears on a large enough even number? So people gradually changed the way they explored problems.
In 1900, Hilbert, the greatest mathematician of the 20th century, listed the "Goldbach Conjecture" as one of 23 mathematical problems at the International Mathematics Conference. Since then, mathematicians of the 20th century "jointly" attacked the fortress of the "Goldbach Conjecture" worldwide, and finally achieved brilliant results.
Proof process
The main methods used by mathematicians in the 20th century to study Goldbach’s conjecture are the sieve method, the circle method, the density method and the triangle sum method. Mathematical method. The idea of solving this conjecture is like "shrinking the encircling circle" and gradually approaching the final result.
In 1920, the Norwegian mathematician Brown proved the theorem "9+9", thus delineating the "big encirclement" for attacking the "Goldbach Conjecture". What is going on with this "9+9"? The so-called "9+9", translated into mathematical language is: "Any even number large enough can be expressed as the sum of the other two numbers, and each of these two numbers is the number of 9 odd prime numbers. Product." Starting from this "9+9", mathematicians all over the world concentrated on "shrinking the encirclement circle". Of course, the final goal is "1+1".
In 1924, German mathematician Redmacher proved the theorem "7+7". Soon, "6+6", "5+5", "4+4" and "3+3" were captured one by one. In 1957, Chinese mathematician Wang Yuan proved "2+3". In 1962, Chinese mathematician Pan Chengdong proved "1+5", and in the same year, he also cooperated with Wang Yuan to prove "1+4". In 1965, Soviet mathematicians proved "1+3".
In 1966, the famous Chinese mathematician Chen Jingrun conquered "1+2", which means: "Any even number large enough can be expressed as the sum of two numbers, and the two numbers One is an odd prime number, and the other is the product of two odd prime numbers." This theorem is called "Chen's Theorem" by the world mathematics community.
Due to Chen Jingrun’s contribution, human beings are only one step away from the final result of Goldbach’s conjecture, "1+1". But in order to achieve this last step, it may have to go through a long process of exploration. Many mathematicians believe that in order to prove "1+1", new mathematical methods must be created, and the past roads are likely to be unworkable.