2. Compatibility of arithmetic axioms The compatibility of Euclidean geometry can be attributed to the compatibility of arithmetic axioms. Hilbert once put forward the proof theory method of formalism plan to prove it. In 193 1, Godel's incompleteness theorem denies this view. 1936 the german mathematician gnc proved the compatibility of arithmetic axioms under the condition of using transfinite induction. The Mathematical Volume of Encyclopedia of China published by 1988 points out that the problem of mathematical compatibility has not been solved.
3. The volumes of two tetrahedrons with equal bases and equal heights are equal. The problem is that there are two tetrahedrons with equal sides and equal heights, which cannot be decomposed into finite small tetrahedrons to make the two groups of tetrahedrons congruent. M.W. Dean gave a positive answer to this question in 1900.
The shortest distance between two points is a straight line. This question is too general. There are many geometries that satisfy this property, so you need to add some restrictions. 1973, the Soviet mathematician Pogrelov announced that this problem was solved under the condition of symmetrical distance. Encyclopedia of China says that after Hilbert, there has been a lot of progress in the construction and discussion of various special metric geometries, but the problems have not been solved.
5. Lie concept of continuous transformation group. The function defining this group is not assumed to be a differentiable problem, but simply called the analyticity of continuous groups, that is, is every local Euclidean group necessarily a Lie group? After von Neumann (1933 for compact groups), Pontryagin (1939 for commutative groups), Shevard (194 1 for solvable groups) and gleason (0952).
6. Axiomatization of physics Hilbert suggested that all physics should be deduced by the axiomatic method of mathematics, first of all probability and mechanics. 1933, the Soviet mathematician Andrei Andrey Kolmogorov realized the axiomatization of probability theory. Later, he achieved great success in quantum mechanics and quantum field theory. However, many people doubt whether physics can be completely axiomatized.
7. The irrational numbers and transcendence of some numbers 1934, A.O. Gelfond and T. Schneider independently solved the latter part of the problem, that is, for any algebraic number α ≠ 0, 1 and any algebraic irrational number β, the transcendence of α β was proved.
8. Prime number problem. Including Riemann conjecture, Goldbach conjecture and twin prime number problem. Riemann conjecture still needs to be solved in general. Goldbach conjectures that the best result belongs to Chen Jingrun (1966), but there is still a long way to go. At present, the best achievement of twin prime number problem also belongs to Chen Jingrun.
9. Prove the most general law of reciprocity in any number field. This problem has been solved by Japanese mathematician Masaji Takagi (192 1) and German mathematician Aiding (1927).
10. Solvability of Diophantine equation. The integer root of an integral coefficient equation can be found, which is called the solvability of Diophantine equation. Hilbert asked, can a general algorithm composed of finite steps be used to judge the solvability of a Diophantine equation? 1970, Ob. Matiyasevich of the Soviet Union proved that the algorithm expected by Hilbert did not exist.
1 1. The coefficient is the quadratic form of any algebraic number. Hasse (1929) and Siegel (1936, 195 1) have obtained important results on this issue.
12. The problem of extending Crocker's theorem in Abelian field to arbitrary algebraic rational number field has only some sporadic results, which is far from being completely solved.
13. It is impossible to solve the ordinary seventh order equation with a function with only two variables. The root of the seventh degree equation depends on three parameters, a, b and c, that is, x=x(a, b, c). Can this function be represented by a binary function? Arnold, a Soviet mathematician, solved the case of continuous function (1957), and Vishkin extended it to the case of continuously differentiable function (1964). But if the requirement is an analytic function, the problem is not solved.
14. Prove the finiteness of a complete function system. This is related to algebraic invariants. 1958, Japanese mathematician Masayoshi Nagata gave a counterexample.
15. The strict basis of Schubert's counting calculus is a typical problem: 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. There are some computable methods that are not closely related to algebraic geometry. But the strict foundation has not been established.
16. The topological problems of algebraic curves and algebraic curves and surfaces are divided into two parts. The first half deals with the maximum number of closed branch curves in algebraic curves. In the second half, we discuss the maximum number and relative position of limit cycles, where X and Y are n-degree polynomials of X and Y. Petrovsky of the Soviet Union claimed to prove that the number of limit cycles does not exceed 3 when n=2, but this conclusion is wrong. Mathematicians in China gave a counterexample (1979).
17. Semi-positive definite representation of sum of squares. For all arrays (x 1, x2, …, xn), polynomials with real coefficients n are always greater than or equal to 0. Can it be written in the form of sum of squares? 1927 Atin proves that this is correct.
18. Construct space with congruent polyhedron. It is partly solved by German mathematicians Biebomach (19 10) and Podinhart (1928).
19. Is the solution of the regular variational problem necessarily analytical? There is little research on this issue. Bernstein and Petrovsky have got some results.
20. The general boundary value problem has developed very rapidly and has become a major branch of mathematics. Research is still going on.
2 1. Proof of the existence of solutions of linear differential equations with single-valued groups. It has been solved by Hilbert himself (1905) and H. Rolle (1957).
22. The univalence of analytic functions composed of automorphism functions. It involves the hard Riemann surface theory. P. Cobb made an important breakthrough in 1907, but other aspects have not been solved.
23. Further development of variational method. This is not a clear mathematical problem, but a general view of variational method. Since the 20th century, the variational method has made great progress.