Goldbach conjecture, for example, and the following examples:
At the 1900 International Congress of Mathematicians held in Paris, Hilbert gave a famous speech entitled "Mathematical Problems". According to the achievements and development trend of mathematical research in the past, especially in the 19th century, he put forward 23 most important mathematical problems. These 23 problems, collectively called Hilbert problems, later became the difficulties that many mathematicians tried to overcome, which had a far-reaching impact on the research and development of modern mathematics and played a positive role in promoting it. Some Hilbert problems have been satisfactorily solved, while others have not yet been solved. The belief that every mathematical problem can be solved in his speech is a great encouragement to mathematicians.
Hilbert's 23 problems belong to four blocks: 1 to 6 are basic mathematical problems; Questions 7 to 12 are number theory problems; Problems 13 to 18 belong to algebraic and geometric problems; 19 to 23 belong to mathematical analysis.
[0 1] Cantor's Continuum Cardinality Problem.
1874, Cantor speculated that there was no other cardinality between countable set cardinality and real set cardinality, that is, the famous continuum hypothesis. 1938, Austrian mathematical logician Godel living in the United States proved that there is no contradiction between the continuum hypothesis and the axiomatic system of ZF set theory. 1963, American mathematician Cohen (p? Choen) proved that the continuum hypothesis and ZF axiom are independent of each other. Therefore, the continuum hypothesis cannot be proved by ZF axiom. In this sense, the problem has been solved.
[02] The contradiction of arithmetic axiom system.
The contradiction of Euclidean geometry can be summed up as the contradiction of arithmetic axioms. Hilbert once put forward the method of proving formalism plan, but Godel's incompleteness theorem published in 193 1 denied it. gnc(g？ Gentaen,1909-1945)1936 proved the non-contradiction of arithmetic axiomatic system by means of transfinite induction.
[03] It is impossible to prove that two tetrahedrons with equal bases and equal heights are equal in volume only according to contract axioms.
The meaning of the problem is: there are two tetrahedrons climbing to the same bottom, which cannot be decomposed into finite small tetrahedrons, so that the two tetrahedrons are congruent (m? Dehn) 1900 has been solved.
The shortest distance between two points is a straight line.
This question is rather general. There are many geometries that satisfy this property, so some restrictions are required. 1973, the Soviet mathematician Bo gref announced that this problem was solved under the condition of symmetrical distance.
[05] Conditions for Topology to Become a Lie Group (Topological Group).
This problem is simply called the analytic property of continuous groups, that is, whether every local Euclidean group must be a Lie group. 1952 was solved by Gleason, Montgomery and Zipin. 1953, Hidehiko Yamanaka of Japan got a completely positive result.
[06] Axiomatization of physics plays an important role in mathematics.
1933, the Soviet mathematician Andrei Andrey Kolmogorov axiomatized probability theory. Later, he succeeded in quantum mechanics and quantum field theory. However, many people have doubts about whether all branches of physics can be fully axiomatized.
[07] Proof of transcendence of some numbers.
Prove: If it is an algebraic number and an algebraic number of irrational numbers, it must be a transcendental number, at least an irrational number (such as sum). Gelfond of the Soviet Union (1929) and Schneider and Siegel of Germany (1935) independently proved its correctness. But the theory of transcendental number is far from complete. At present, there is no unified method to determine whether a given number exceeds the number.
[08] The problem of prime number distribution, especially for Riemann conjecture, Goldbach conjecture and twin prime numbers.
Prime number is a very old research field. Hilbert mentioned Riemann conjecture, Goldbach conjecture and twin prime numbers here. Riemann conjecture is still unsolved. Goldbach conjecture and twin prime numbers have not been finally solved, and the best result belongs to China mathematician Chen Jingrun.
[09] Proof of general reciprocity law in arbitrary number field.
192 1 year was ruled by takagi of Japan, 1927 was ruled by Anting of Germany (e? Artin) Give each other a basic solution. However, category theory is still developing.
[10] Can we judge whether the indefinite equation has a rational integer solution by finite steps?
Finding the integer root of the integral coefficient equation is called Diophantine (about 2 10-290, an ancient Greek mathematician) equation solvable. Around 1950, American mathematicians such as Davis, Putnam and Robinson made key breakthroughs. In 1970, Baker and Feros made positive conclusions about the equation with two unknowns. 1970. The Soviet mathematician Marty Sevic finally proved that, on the whole, the answer is negative. Although the result is negative, it has produced a series of valuable by-products, many of which are closely related to computer science.
The theory of quadratic form in [1 1] algebraic number field.
German mathematicians Hassel and Siegel made important achievements in the 1920s. In the 1960s, the French mathematician Wei Yi (A? Weil) has made new progress.
The composition of [12] class domain.
That is, Kroneck's theorem on Abelian field is extended to any algebraic rational field. This problem has only some sporadic results and is far from being completely solved.
[13] The impossibility of solving the general algebraic equation of degree seven by the combination of binary continuous functions.
The root of the seventh-order equation depends on three parameters in the equation,,; . Can this function be represented by a binary function? This problem is about to be solved. 1957, the Soviet mathematician Arnold proved that any continuous real function in the world can be written in form, in which sum is a continuous real function. Andre Andrey Kolmogorov's proof can be written in form, in which sum is a continuous real function, and the choice can be completely irrelevant. In 1964, Vituskin is extended to the case of continuously differentiable, but the case of analytic function is not solved.
[14] The finite proof of some complete function systems.
That is, polynomials with independent variables over a field are rings of rational functions over a field. Can they be generated by polynomials with finite elements? In 1959, Masayoshi Nagata, a Japanese mathematician, gave a negative solution to this problem related to algebraic invariants with a beautiful counterexample.
[15] Establish the foundation of algebraic geometry.
Dutch mathematicians Vander Waals Deng 1938 to 1940 and Wei Yi 1950 have solved the problem.
Note: Schubert's strict foundation of counting calculus.
A typical problem is that there are four straight lines in three-dimensional space. How many straight lines can intersect all four? Schubert gave an intuitive solution. Hilbert asked to generalize the problem and give a strict basis. Now there are some computable methods, which are closely related to algebraic geometry. But the strict foundation has not been established.
Topological research on [16] algebraic curves and surfaces.
The first half of this problem involves the maximum number of closed bifurcation curves in algebraic curves. In the second half, it is required to discuss the maximum number and relative position of spare limit cycles, where is, is a polynomial. For the case of the secondary system, 1934 Fu Luoxian got it; Bao Ting got it on 1952; 1955, Podlovschi of the Soviet Union announced that this result, which had been shaken for some time, was in doubt because some lemmas were rejected. Regarding the relative position, China mathematician and Ye proved in 1957 that there are no more than two strings. In 1957, China mathematicians Qin Yuanxun and Pu Fujin illustrated that the equation has at least three series limit cycles. In 1978, under the guidance of Qin Yuanxun and Hua, Shi Songling and Wang of China respectively gave at least four concrete examples of limit cycles. In 1983, Qin Yuanxun further proved that the quadratic system has at most four limit cycles and is a structure, thus finally solving the structural problem of the solution of the quadratic differential equation and providing a new way to study the Hilbert problem [16].
The square sum representation of [17] semi-positive stereotypes.
For any array, the rational function of real coefficients is always greater than or equal to 0. Are they all written as the sum of squares of rational functions? 1927 Atin has been definitely solved.
[18] Construct space with congruent polyhedron.
German mathematicians Bieber Bach (19 10) and Reinhardt (1928) gave some answers.
[19] Is the solution of the regular variational problem always an analytic function?
German mathematician Berndt (1929) and Soviet mathematician Petrovsky (1939) have solved this problem.
[20] Study the general boundary value problem.
This problem is progressing rapidly and has become a major branch of mathematics. I was still researching and developing a few days ago.
[2 1] Proof of the existence of solutions for Fuchs-like linear differential equations with given singularities and single-valued groups.
This problem belongs to the large-scale theory of linear ordinary differential equations. Hilbert himself is in 1905, Loer (h? Rohrl) obtained important results in 1957 respectively. Deligne, a French mathematician from 65438 to 0970, made outstanding contributions.
[22] Analytic function is univalued by automorphism function.
This problem involves the difficult Riemannian surface theory, 1907 Keber (P? Koebe) solved a variable situation and made an important breakthrough in the research of the problem. Other aspects have not been solved.
[23] to carry out the study of variational method.
This is not a clear mathematical problem. Variational method has made great progress in the 20th century.