[汇集，科普，数学] 1900年希尔伯特的 23个问题（英文版，简版）

一、1902年，美国数学会

   David Hilbert. Mathematical problems[J]. Bulletin of the American Mathematical Society, 1902, 8(10): 437-479.

   doi:  10.1090/S0002-9904-1902-00923-3

1. Cantor's problem of the cardinal number of the continuum.

2. The compatibility of the arithmetical axioms.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. Problem of the straight line as the shortest distance between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

6. Mathematical treatment of the axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Problems of prime numbers (The "Riemann Hypothesis").

9. Proof of the most general law of reciprocity in any number field.

10. Determination of the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic numerical coefficients.

12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality.

13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.

14. Proof of the finiteness of certain complete systems of functions.

15. Rigorous foundation of Schubert's enumerative calculus.

16. Problem of the topology of algebraic curves and surfaces.

17. Expression of definite forms by squares.

18. Building up of space from congruent polyhedra.

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

20. The general problem of boundary values. (Boundary value problems in PD)

21. Proof of the existence of linear differential equations having a prescribed monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. Further development of the methods of the calculus of variations.

二、欧洲数学会， Encyclopedia of Mathematics ，2018

1. Cantor's problem on the cardinal number of the continuum.

2. The compatibility of the arithmetical axioms.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. The problem of the straight line as the shortest distance between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

6. Mathematical treatment of the axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Problems of prime numbers (cf. also Prime number).

9. Proof of the most general law of reciprocity in any number field.

10. Determination of the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic numerical coefficients.

12. Extension of the Kronecker theorem on Abelian fields to any algebraic realm of rationality.

13. Impossibility of the solution of the general equation of the 7

-th degree by means of functions of only two variables.

14. Proof of the finiteness of certain complete systems of functions.

15. Rigorous foundation of Schubert's enumerative calculus.

16. Problem of the topology of algebraic curves and surfaces.

17. Expression of definite forms by squares.

18. Building up of space from congruent polyhedra.

19. Are the solutions of the regular problems in the calculus of variations always necessarily analytic.

20. The general problem of boundary values.

21. Proof of the existence of linear differential equations having a prescribed monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. Further development of the methods of the calculus of variations.

三、苏格兰，《 MacTutor 》版，2014

1. The cardinality of the continuum, including well-ordering.

2. The consistency of the axioms of arithmetic.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. The straight line as shortest connection between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining a group.

6. The axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Prime number theorems (including the Riemann hypothesis).

9. The proof of the most general reciprocity law in arbitrary number fields.

10. Decision on the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic coefficients.

12. The extension of Kronecker's theorem on Abelian fields to arbitrary algebraic fields.

13. Impossibility of solving the general seventh degree equation by means of functions of only two variables.

14. Finiteness of systems of relative integral functions.

15. A rigorous foundation of Schubert's enumerative calculus.

16. Topology of real algebraic curves and surfaces.

17. Representation of definite forms by squares.

18. The building up of space from congruent polyhedra.

19. The analytic character of solutions of variation problems.

20. General boundary value problems.

21. Linear differential equations with a given monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. The further development of the methods of the calculus of variations.

[24.] The simplicity of proofs (omitted).

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