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.

(1) Cardinality of Cantor continuum.

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 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.

(2) Arithmetic axiom system is not contradictory.

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. genta en,1909-1945)1936 proved the non-contradiction of the arithmetic axiomatic system by means of transfinite induction.

(3) It is impossible to prove that two tetrahedrons with equal base and equal height are equal in volume only according to the contract axiom.

The significance of the problem is that there are two tetrahedrons with equal height, which cannot be decomposed into finite small tetrahedrons, so that the congruence of the two tetrahedrons (M. DEHN) has been solved in 1900.

(4) Take a straight line as the shortest distance between two points.

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.

(5) Conditions for topology to be 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.

(6) Axiomatization of physics, which 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.

(7) Proof of transcendence of some numbers.

It is proved that if α is algebraic number and β is algebraic number of irrational number, then α β must be transcendental number or at least irrational number (for example, 2√2 and eπ). 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.

(8) The distribution of prime numbers, 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.

(9) Proof of the general law of reciprocity in arbitrary number field.

192 1 was basically solved by Kenji Takagi of Japan, and 1927 was basically solved by E.Artin of Germany. However, category theory is still developing.

(10) Can we judge whether an 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.

Quadratic theory in (1 1) algebraic number field.

German mathematicians Hassel and Siegel made important achievements in the 1920s. In 1960s, French mathematician A.Weil made new progress.

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.

The impossibility of (13) combination of binary continuous functions to solve the seventh general algebraic equation.

The root of equation x7+ax3+bx2+cx+ 1=0 depends on three parameters A, B and C; X=x(a, b, c). Can this function be represented by a binary function? This problem is about to be solved. 1957 Arnold, a Soviet mathematician, proved that any continuous real function f(x 1, x2, x3) on [0, 1] can be written in the form of ∑ hi (ξi (x 1, x2), x3) (i. X3) can be written as ∑ hi (ξ i 1 (x 1) in 1964. Vituskin is extended to continuously differentiable, but the analytic function is not solved.

The finite proof of (14) some complete function systems.

That is, the polynomial fi (I = 1, ..., Xn), where r is the negative solution of this problem related to algebraic invariants by the rational function F(X 1, ..., Xm) and F. Japanese mathematician Masayoshi Nagata in 1959.

(15) Establish the foundation of algebraic geometry.

Dutch mathematicians Vander Waals Deng 1938 to 1940 and Wei Yi 1950 have solved the problem.

Note 1 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 N(n) and relative position of limit cycles of dx/dy=Y/X, where x and y are polynomials of degree n of x and y. For the case of n=2 (i.e. quadratic system), 1934, Froxianer obtains n (2) ≥1; 1952, Bao Ting got n (2) ≥ 3; 1955, Podlovschi of the Soviet Union declared that n(2)≤3, which was the result of a shock for a while, but was questioned because some lemmas were rejected. Regarding the relative position, China mathematician and Ye proved in 1957 that (E2) does not exceed two strings. In 1957, China mathematicians Qin Yuanxun and Pu Fujin gave a concrete example. The equation with n = 2 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 the structure is (1 3), thus finally solving the structural problem of the solution of the quadratic differential equation and providing a new way for studying the Hilbert problem (16).

The square sum representation of (17) semi-positive definite form.

The rational function f (x 1, ..., xn) is for any array (x 1, ..., xn). Are you sure that f can be 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 obtained important results in 1905 and H.Rohrl in 1957 respectively. Deligne, a French mathematician from 65438 to 0970, made outstanding contributions.

(22) Automorphic single-valued analytic function.

This problem involves the difficult Riemann surface theory. In 1907, P.Koebe solved a variant and made an important breakthrough in the study of this problem. Other aspects have not been solved.

(23) Carry out the research of variational method.

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