Mathematics (English: mathematics; derived from the ancient Greek language μ ≤θη μ α, m á th e ma;Abbreviated as math or math), it is a door to study the concepts of quantity, structure, change, space and informationsubject。
Mathematics is a universal means for humans to strictly describe and deduce the abstract structure and pattern of things. It can be applied to any problem in the real world. All mathematical objects are artificially defined in essence.In this sense, mathematics belongs toFormal science, notnatural science。Different mathematicians and philosophers have a series of views on the exact scope and definition of mathematics.
Mathematics plays an irreplaceable role in the development of human history and social life, and it is also an essential basic tool for learning and studying modern science and technology.
a: Deductionlogic(also called symbolic logic), b: proof theory (also called meta mathematics), c: recursion theory, d: model theory, e: axiomatic set theory, f: mathematical foundation, g: mathematical logic and mathematical foundation other disciplines.
a: Elementary number theory, b: analytic number theory, c: algebraic number theory, d: transcendental number theory, e: Diophantine approximation, f: geometry of numbers, g: probabilistic number theory, h: computational number theory, i: number theory and other disciplines.
a: Linear algebra, b: group theory, c: field theory, d: Lie group, e: Lie algebra, f: Kac Moody algebra, g: ring theory (including commutative ring and commutative algebra, associative ring and associative algebra, non associative ring and non associative algebra, etc.), h: module theory, i: lattice theory, j: universal algebra theory, k: category theory, l: homology algebra, m: algebraic K theory, n: differential algebra,o: Algebraic coding theory, p: algebra and other disciplines.
a: Real variable function theory, b: single complex variable function theory, c: multiple complex variable function theory, d: function approximation theory, e: harmonic analysis, f: complex manifold, g: special function theory, h: function theory and other disciplines.
a: Geometric probability, b: probability distribution, c: limit theory, d: stochastic process (including normal process, stationary process, point process, etc.), e: Markov process, f: stochastic analysis, g: martingale theory, h: applied probability theory (specifically applied to relevant disciplines), i: probability theory and other disciplines.
twenty-sixapplied mathematics(Specific application into relevant disciplines)
27. Other Mathematics Disciplines
Development history
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Mathematics（Chinese Pinyin：shù xué；Greek: μ α θη μ α τ τ κ;English: mathematics or math), whose English is derived fromAncient GreekWith learningknowledgeThe meaning of science.ancient GreekScholars regard it as the starting point of philosophy and the "foundation of learning".In addition, there is a narrow and technical meaning -“Mathematical research”。Even in its etymology, its adjective meaning, where relevant to learning, is also used to refer to mathematics.
Its EnglishcomplexThe form, and the plural form in French plus es, into math é matiques, which can be traced back to Latinneuter plural (mathematica), byCiceroTranslated from the Greek complex number τα μ α θημ α τ κ ≤ (ta math e matik á).
In ancient China, mathematics was called arithmeticArithmetics, and finally changed to mathematics.Arithmetic in ancient China is one of the six arts (called "number" in the six arts).
Mathematics originated from early human production activities,Babylon, CubaPeople fromremote dateAt the beginning, I have accumulated some mathematical knowledge and can apply practical problems.From the perspective of mathematics itself, their mathematical knowledge is only the result of observation and experience, without comprehensive conclusions and proofs, but their contributions to mathematics should also be fully recognized.
Basic MathematicsThe application of knowledge is an indispensable part of individual and group life.Its basic concepts were refined as early as in ancient EgyptMesopotamiaandancient IndiaInternalAncient mathematicsYou can see it in the text.Since then, its development has continued to make small progress.But algebra and geometry at that time remained independent for a long time.
Algebra can be said to be the most widely accepted "mathematics".It can be said that since everyone started to learn to count when he was young, the first mathematics he came into contact with was algebra.Mathematics, as a discipline studying "numbers", algebra is also one of the most important parts of mathematics.Geometry is the first branch of mathematics to be studied.
untilsixteenth centuryOfThe RenaissancePeriod,Descartes Created analytic geometry, which completely separatedAlgebraAnd geometry.Since then, we can finally prove the theorems of geometry by calculation;At the same time, abstractalgebraic equationAnd trigonometric functions.Later, more subtleCalculus。
Mathematics is used in many differentfieldUp, includingscience、engineering、Medical ScienceandeconomicsEtc.The application of mathematics in these fields is generally called applied mathematics, and sometimes it will stimulate new mathematical discoveries and promote the development of new mathematical disciplines.Mathematicians also study pure mathematics, that is, mathematics itself, without taking any practical application as the goal.Although there is a lot of work to study pure mathematics as a beginning, but later may find appropriate applications.
Specifically, there are sub fields used to explore the connection between the core of mathematics and other fields: logic, set theory（Fundamentals of Mathematics）, to empirical mathematics of different sciences (applied mathematics)UncertaintyResearch on（chaos、fuzzy mathematics）。
As far as the vertical dimension is concerned, the exploration in the respective fields of mathematics is also getting deeper and deeper.
definition
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Aristotle defined mathematics as "quantitative mathematics" until the 18th century.Since the 19th century, mathematical research has become more and more rigorous, involving abstract topics such as group theory and projection geometry that have no clear relationship with quantity and measurement. Mathematicians and philosophers have begun to put forward various new definitions.Some of these definitions emphasize the deductive nature of a large number of mathematics, some emphasize its abstraction, and some emphasize some topics in mathematics.Even among professionals, there is no consensus on the definition of mathematics.There is even no consensus on whether mathematics is art or science.[8] Many professional mathematicians are not interested in the definition of mathematics, or think it is undefinable.Some just say, "Mathematicians do math."
The three main types of mathematical definitions are called logicians, intuitionists and formalists, each reflecting different philosophical schools of thought.There are serious problems, which are not universally accepted, and no reconciliation seems feasible.
The early definition of mathematical logic is Benjamin Peirce's "science of drawing necessary conclusions" (1870).In Principia Mathematica, Bertrand Russell and Alfred North Whitehead put forward a philosophical program called logicism, and tried to prove that all mathematical concepts, statements and principles can be defined and proved by symbolic logic.The logical definition of mathematics is Russell's "all mathematics is symbolic logic" (1903).
Intuitionism definition, from mathematician LE. J. Brouwer，Identify mathematics with certain mental phenomena.An example of the definition of intuitionism is "mathematics is a psychological activity that is constructed one after another".Intuitionism is characterized by its rejection of some mathematical ideas that are considered valid according to other definitions.In particular, although other mathematical philosophy allows objects that can be proved to exist, even though they cannot be constructed, intuitionism only allows mathematical objects that can be actually constructed.
Formalism defines mathematics with its symbols and operating rules.Haskell Curry simply defines mathematics as "the science of formal systems".[33] The formal system is a group of symbols or tokens. There are also some rules that tell tokens how to combine into formulas.In the formal system, the word axiom has a special meaning, which is different from the common meaning of "self-evident truth".In a formal system, an axiom is a combination of tokens contained in a given formal system, without the need to use the system's rule export.[1]
structure
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Many such as numbersfunction、geometryThe mathematical objects of, etc. reflect the internal structure of continuous operations or relations defined therein.Mathematics studies the properties of these structures. For example, number theory studies how integers are represented in arithmetic operations.In addition, different structures have similar properties, which makes it possible to describe their states by using axioms for a class of structures through further abstraction. What needs to be studied is to find structures that meet these axioms in all structures.So we can learngroup, rings, domains, and other abstract systems.These studies (through the structure defined by algebraic operations) can form the field of abstract algebra.Because abstract algebra has great versatility, it can often be applied to some seemingly unrelated problems, such as some ancient drawing problems with rulers and compassesGalois theoryYes, it involvesField theoryandgroup theory。Another example of algebraic theory is linear algebra, which has quantitative and directional properties for its elementsvector space A general study has been made.These phenomena indicate that geometry and algebra, which were previously considered uncorrelated, actually have strong correlations.Combinatorics studies the method of enumerating a number of objects that satisfy a given structure.
Number and space play an important role in analytic geometry, differential geometry and algebraic geometry.In differential geometryFibre plexusandmanifoldThe concept of computing on.In algebraic geometry, there arepolynomialEquationalSolution setThe description of geometric objects, such as, combines the concepts of number and space;There are alsoTopological groupThe study of the structure and space.Lie groups are used to study space, structure and change.
In order to understand the basis of mathematics, mathematical logic and set theory have been developed.German mathematiciancantor (1845-1918) pioneered set theory, boldly“Infinity”The purpose of the march is to provide a solid foundation for all branches of mathematics, and its content is also quite richReal infinityHis thought has made immeasurable contributions to the development of mathematics in the future.
At the beginning of the 20th century, set theory has gradually penetrated into various branches of mathematicsAnalytical theoryMeasurement theoryIt is an indispensable tool in topology and mathematical science.At the beginning of the 20th century, mathematician Hilbert spread Cantor's thought in Germany, calling set theory "mathematician's paradise" and "the most amazing product of mathematical thought".Russell, the British philosopher, praised Cantor's work as "the greatest work that can be boasted of in this era".
Mathematical logic focuses on putting mathematics on a solid axiom framework and studying the results of this framework.In itself, it is the origin of the second incomplete theorem of Godel, which is perhaps the most popular achievement in logic. Modern logic is divided intoRecursion theory、Model theoryandProof theory, andTheoretical Computer ScienceIt is closely related.
Maybe in ancient ChinaCalculationIt is one of the earliest symbols in the world, originated fromShang dynastyOur divination.
Most of what we use todayMathematical symbolIt was not invented until the 16th century.Before that, mathematics was written in words, which is a hard program that will limit the development of mathematics.Today's symbols make mathematics easier for people to operate, but beginners often feel shy about it.It is extremely compressed: a small number of symbols contain a large amount of information.Like musical symbols, today's mathematical symbols have clear syntax and encoding of information that is difficult to write in other ways.
rigorism
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Math language is also difficult for beginners.How to make these words have more precise meanings than ordinary words also troubles beginners, such as opening andfieldThe word "etc." has a special meaning in mathematics.Mathematical terminologyIt also includesHomeomorphismAnd integrability.But there is a reason to use these special symbols and terminology: mathematics needs more precision than ordinary language.Mathematicians call this requirement for language and logical accuracy "preciseness".
Mathematics is a universal means for humans to strictly describe the abstract structure and pattern of things, and can be applied to any problem in the real world.In this sense, mathematics belongs toFormal science, not natural science.All mathematical objects are artificially defined in essence. They do not exist in nature, but only in human thinking and concepts.As a result, the correctness of mathematical propositions cannot be tested by repeated experiments, observations or measurements, as physics, chemistry and other natural sciences with the goal of studying natural phenomena, but directly by rigorous logicreasoningTo prove it.Once the conclusion is proved by logical reasoning, the conclusion is also correct.
Mathematicalaxiomatic method In essence, it is the direct application of logical methods in mathematics.In the axiomatic system, allpropositionIt is connected with proposition by rigorous logic.Directly adopted without definitionPrimitive conceptStarting, other derived concepts are gradually established by means of logical definition;Starting from the axiom that is directly used as the premise without proof, further conclusions are gradually drawn by means of logical deduction, that istheorem；Then all the concepts and theorems are combined into a whole with internal logical links, which constitutes an axiom system.
Rigidity is an important and basic part of mathematical proof.Mathematicians hope that their theorems will be deduced according to axioms by systematic reasoning.This is to avoid getting wrong "theorems" or "proofs" by relying on unreliable intuition, which has seen many examples in history.The preciseness expected in mathematics varies with time: the Greeks expected careful arguments, but in Newton's time, the methods used were less rigorous.Newton's definition to solve the problem has arrivednineteenth centuryLet mathematicians use rigorous analysis and formal proof to properly handle.Mathematicians continue to argue about the rigour of computer-aided proof.When a large number of calculations are difficult to verify, it is difficult to say that the proof is effective and rigorous.
number
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The study of quantity starts from numbers, starting with familiar natural numbers and integers and rational and irrational numbers described in arithmetic.
To be specific, because of the need to count, human beings have abstracted natural numbers from real things, which is the starting point of all "numbers" in mathematics.Natural numbers are not closed to subtraction. In order to close subtraction, we expand the number system to integers;In order not to close division, but to close division, we extend the number system to rational numbers;For the open square operation, we extend the number system to algebraic numbers (in fact, algebraic numbers are a broader concept).On the other hand, for the limit operation is not closed, we extend the number system to real numbers.Finally, we extend the number system to complex numbers in order to avoid that negative numbers cannot open even power operations in the real number range.Complex number is the smallest algebraic closed field containing real number. We conduct four operations on any complex number, and the reduction results are all complex numbers.
Another concept related to "quantity" is the "potential" of infinite sets, which leads to another concept of infinity: Alev number, which allows meaningful comparisons between the sizes of infinite sets.
brief history
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A Brief History of Western Mathematics
The evolution of mathematics can be seen asAbstractionEuropean civilization developed geometry, while China developed arithmetic.The first concept to be abstracted is probably the number (Chinese arithmetic). Its cognition of the same thing between two apples and two oranges is a breakthrough in human thought. In addition to the cognition of how to count the number of actual objects, prehistoric humans also understood how to count the number of abstract concepts, such as time - day, season and year.Arithmetic (addition, subtraction, multiplication and division) also naturally came into being.
Further, writing or other digital recording systems are required, such asRune woodOr onIncaKip used.There have been many different counting systems in history.
In ancient times, mathematicalmainThe principle is to study astronomy, landgrain cropsReasonable distribution, tax and trade related calculations.Mathematics is formed to understand the relationship between numbers, measure the land and predict astronomical events.These needs can be simply summarized as mathematical research on quantity, structure, space and time.
The emergence of the concept of variable in Europe in the 17th century made people begin to study the relationship between quantity and quantity in change and the mutual transformation between figures.During the establishment of classical mechanicsCalculusThe method was invented.With the further development of natural science and technology, set theory andmathematical logicAnd other fields began to develop slowly.[2]
Mathematics, known as mathematics in ancient times, was an important discipline in ancient Chinese science. According to the characteristics of the development of ancient Chinese mathematics, it can be divided into five periods: germination;The formation of the system;development;Prosperity and the integration of Chinese and Western mathematics.
relevant
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Many research achievements in ancient Chinese arithmetic have already bred the thinking methods that were later involved in western mathematics. In modern times, many world leading mathematical research achievements are named after Chinese mathematicians:
【Gardenvale coordinate】mathematicianZhou WeiliangThe research achievements in algebraic geometry are called“Zhou's coordinate；In addition, there is one named after him“Zhou's theorem”And“Zhou's ring”。
【Wang'sParadox]A proposition of mathematician Wang Hao on mathematical logic is internationally recognized as“Wang's paradox”。
【Coriolis theorem】The research achievements of mathematician Ke Zhao on Cartland problem are called by the international mathematical community“Coriolis theorem”；In addition, his research achievements in number theory with mathematician Sun Qi are internationally known as“Ke Sun conjecture”。
【Chen's theorem 】Mathematician Chen JingrunGoldbach conjectureThe proposition proposed in the research is praised by the international mathematical community“Chen's theorem ”。
【Yang Zhang Theorem】Mathematician Yang LeheZhang GuanghouResearch achievements in function theory are internationally known as“Yang Zhang Theorem”。
【Lu's conjecture】mathematicianLu QikengThe research results on constant curvature manifolds are internationally known as“Lu's conjecture”。
[Xia's inequality]mathematicianSummer WalkThe research achievements in functional integral and invariant measure theory are called“Xia's inequality”。
【Jiang's space】mathematicianJiang BojuThe research results on the calculation of Nielsen number are internationally named“Jiang's space”；In addition, there is one named after him“Jiang's subgroup”。
【Wang's theorem】mathematicianWang XutangThe research achievements on point set topology are praised by the international mathematical community“Wang's theorem”。
【Yuan's lemma】mathematicianYuan YaxiangThe research achievements in nonlinear programming have been internationally named“Yuan's lemma”。
【King operator】mathematicianJing NaihuanThe research achievements in symmetric functions are internationally named“Jingoperator”。
【Chen's Grammar】mathematicianChen YongchuanIn groupComposite numberThe research achievements in science have been internationally named“Chen's Grammar”。
Mathematics is the word God uses to write the universe——Galileo
I am determined to give up that only abstract geometry.That is to say, I will no longer consider the problems that are only used to practice thinking. I do so to study another kind of geometry, that is, the geometry aimed at explaining natural phenomena——Descartes（Rene Descartes，1596 ~ 1650）
Mathematicians are trying to discover some order of prime number sequence on this day. We have reason to believe that this is a mystery, and the human mind can never penetrate——Euler
Some beautiful theorems in mathematics have such characteristics: they are very easy to be induced from facts, but the proofs are extremely hidden.Mathematics is the king of science——Gaussian
This is the advantage of well structured language. Its simplified notation is often the source of profound theories——Laplace（Pierre Simon Laplace，1749 ~ 1827）
If you think that onlyGeometric proofIt would be a serious mistake if there was necessity in the evidence of feeling——Cauchy（Augustin Louis Cauchy，1789 ~ 1857）
The essence of mathematics lies in its freedom——cantor （Georg Ferdinand Ludwig Philipp Cantor，1845 ~ 1918）
Music can stimulate or comfort feelings, painting can make people happy, poetry can move people's hearts, philosophy can make people get wisdom, science can improve material life, but mathematics can give all the above——Christian Felix Klein (1849~1925)
As long as a branch of science can raise a large number of questions, it is full of vitality, and the lack of questions indicates the termination or decline of independent development——David Hilbert (1862-1943)
Problems are the heart of mathematics——Paul Halmos（Paul Halmos，1916 ~ 2006）
Time is a constant, but for diligent people, it is a "variable".People who use "minute" to calculate time spend 59 times more time than people who use "hour" to calculate time——Rybakov
Chinese characters
There are different kinds of things, and each has its own way. So although the branches are divided into different parts, they can know the same thing, but only one end of the branch is distributed. The reason is explained in words, and when the map is broken up, the common people can also agree with each other, but they can understand each other without being militaristic. Those who look at it think too much——Liu Hui
The rate of late illness is not a supernatural monster. It is visible and can be detected. There are several that can be inferred. -- Zu Chongzhi (429~500)
Mathematics is accurate and concise in expression, abstract and universal in logic, and flexible in form. It is an ideal tool for universal communication——Zhou Haizhong
Science needs experiments, but experiments cannot be absolutely accurate. If there is a mathematical theory, it is completely correct to rely on inference. This science cannot leave the reason of mathematics
Many basic scientific concepts often need mathematical concepts to express them. So mathematicians have enough to eat, but they can'tNobel PrizeIt is natural. There is no Nobel Prize in mathematics, which may be a good thing. Nobel Prize is too eye-catching, which will make mathematicians unable to focus on their own research——Chen Xingshen
After modern high-energy physics arrived at quantum physics, many of them could not do experiments at all. They were calculated with pen and paper at home, which was not far from what mathematicians thought. Therefore, mathematics has an incredible power in physics——Qiu Chengtong
We should pay attention to the order of reading and writing homework. We should develop good learning methods. We should try to review the knowledge we learned that day after we go home. Especially, we should pay attention to the notes we have written, and then write homework. This will have a better effect
punctuation
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Mathematics is an international discipline, which requires preciseness in all aspects.
Mathematics at or above the elementary level stipulated by China can be regarded as scientific and technological literature.
China stipulates that the full stop of literature articles must use "." The purpose of mathematical use is to avoid confusion with subscripts. The third reason is that China has submitted mathematical research reports internationally, but others do not use them, because most foreign full stops are not "."
In the proof question, ∨ (because) should be followed by ",", ∨ (so) should be followed by ".". If there are several small questions in a big question, ";" should be followed at the end of each small question, "." should be followed at the end of the last question, and ";" should be followed at the end of such serial numbers as ① ② ③ ④, and "." should be followed at the end of the last serial number
Discipline distribution
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National key discipline with first-class mathematics disciplineUniversity:
The first seven problems are recognized as the seven problems, and the eighth problem is one of the three guesses in the world.
1、 P (polynomial algorithm) problem versus NP (non polynomial algorithm) problem
On a Saturday night, you attended a grand party.Feeling embarrassed, you want to know if there are people you already know in this hall.Your host suggests to you that you must know the lady Rose who is in the corner near the dessert plate.It doesn't take you a second to scan there and find that your master is right.However, if there is no such hint, you must look around the hall and examine everyone one by one to see if there are people you know.It usually takes much more time to generate a solution to a problem than to verify a given solution.This is an example of this general phenomenon.
Similarly, if someone tells you that the number 13717421 can be written as the product of two smaller numbers, you may not know whether you should trust him or not, but if he tells you that it can be factored into 3607 times 3803, then you can easily verify this with a pocket calculator.No matter whether we are clever in writing programs, it is considered as one of the most prominent problems in logic and computer science to determine whether an answer can be quickly verified by using internal knowledge, or it takes a lot of time to solve without such prompts.It isSteven·Stephen Cook made a statement in 1971.
2、 Hodge conjecture
Mathematicians in the twentieth century found a powerful way to study the shape of complex objects.The basic idea is to ask to what extent we can change the shape of a given object bydimensionAn increasing number of simple geometric building blocks are glued together to form.This technique has become so useful that it can be promoted in many different ways;Finally, it leads to some powerful tools that enable mathematicians to make great progress in classifying various objects they encounter in their research.Unfortunately, in this generalization, the geometric starting point of the program becomes blurred.In a sense, some parts without any geometric explanation must be added.Hodge conjecture Assertions, for the so-calledProjective algebraic varietyFor this particularly perfect space type, the part called Hodge closed chain is actually calledAlgebraic closed chainA (rational linear) combination of geometric components of.
3、 Poincare conjecture (proved)
If we stretch the rubber band around the surface of an apple, we can neither break it nor let it leave the surface, so that it slowly moves and shrinks to a point.On the other hand, if we imagine that the same rubber belt is telescoped on a tire surface in an appropriate direction, we cannot shrink it to a point without breaking the rubber belt or tire surface.We say that the surface of the apple is“simply-connected And the tire tread is not.About a hundred years ago, Poincare had known that the two-dimensional sphere could be characterized by simple connectivity in essence, and he proposedThree-dimensional sphere（Four-dimensional space)The corresponding problem of all points with unit distance from the origin in.The problem immediately became extremely difficult. Since then, mathematicians have been struggling for it.
4、 Riemann hypothesis
Some numbers have special properties that cannot be expressed as the product of two smaller numbers, such as 2, 3, 5, 7, etc.Such numbers are called prime numbers;They play an important role in pure mathematics and its application.At allNatural numberThe distribution of such prime numbers does not follow any regular pattern;However, German mathematician Riemann (1826~1866) observed that the frequency of prime numbers is closely related to the behavior of an elaborately constructed so-called Riemann Zetta function z (s).famousRiemann hypothesisIt is asserted that all meaningful solutions of equation z (s)=0 are on a straight line.This has been verified for the first 150000000 solutions.To prove that it holds true for every meaningful solution will bePrime distributionMany mysteries of bring light.
5、 Yang Mills existence and quality gap
The laws of quantum physics are based on classical mechanicsNewton's lawThe way to the macro worldElementary particleThe world was founded.About half a century ago,Yang ZhenningAnd Mills found that quantum physics reveals the striking relationship between elementary particle physics and the mathematics of geometric objects.The prediction based on Young Mills equation has been confirmed in the following high-energy experiments performed in laboratories around the world: BrockHaven、Stanford、European Institute for Particle PhysicsandTsukuba。However, their equations that describe heavy particles and are mathematically rigorous have no known solutions.In particular, it is confirmed by most physicists, and“quark”The "quality gap" hypothesis applied in the interpretation of invisibility has never been proved mathematically satisfactory.Progress on this issue requires the introduction of fundamental new ideas in physics and mathematics.
6、 Existence and Smoothness of Navier Stokes Equations
The undulating waves follow our boat winding through the lake, and the turbulent current follows our modernityJet aircraftThe flight of.Mathematicians and physicists are convinced that both breeze and turbulence can be explained and predicted by understanding the solution of Navier Stokes equation.Although these equations were written in the 19th century, we still have very little understanding of them.The challenge is to make substantial progress in mathematical theory so that we can solve the mystery hidden in the Navier Stokes equation.
7、 Birch and Swinnerton Dyer conjectures
Mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as x ^ 2+y ^ 2=z ^ 2.EuclidThe complete solution of this equation was once given, but for more complex equations, it becomes extremely difficult.In fact, as Yu V. Matiyasevich pointed out, Hilbert's tenth problem is unsolvable, that is, there is no general method to determine whether such a method has an integer solution.When the solution is a point of an Abelian cluster, the Bach and Swinton Dale conjectures hold that,Rational pointThe size of the group of is related to the behavior of a Zetta function z (s) near the point s=1.In particular, this interesting conjecture holds that if z (1) is equal to 0, then there are infinite rational points (solutions);On the contrary, if z (1) is not equal to 0, there are only a limited number of such points.
8、 Goldbach conjecture
In his letter to Euler on June 7, 1742, Goldbach put forward the following conjecture: (a) Any even number not less than 6 can be expressed as twoOdd prime numberSum of;(b) Any odd number not less than 9 can be expressed as the sum of three odd prime numbers.Euler also proposed another equivalent version in his reply, that is, any even number greater than 2 can be written as twoPrime numberThe sum of.These two propositions are generally called Goldbach conjecture.The proposition "any large even number can be expressed as the sum of a number with no more than a prime factor and another number with no more than b prime factors" is recorded as "a+b",Coriolis The guess is to prove that "1+1" is true.
In 1966, Chen Jingrun proved the establishment of "1+2", that is, "any large even number can be expressed as the sum of a prime number and another prime factor that does not exceed 2".[3]
