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Hilbert's Mathematical Problems Table of contents (The actual text is on a separate page.) (Philosophy of problems, relationship between mathematics and science, role of Introduction. proofs, axioms and formalism.) Treatise print. Problem 1. Cantor's problem of the cardinal number K. Gödel. The consistency of the of the continuum. (The continuum axiom of choice and of the hypothesis.) Treatise print. generalized continuum hypothesis. Princeton Univ. Press, Princeton, 1940. Problem 2. The compatibility of the arithmetical axioms. Treatise print. Problem 3. The equality of two volumes of two V. G. Boltianskii. Hilbert's Third tetrahedra of equal bases and equal Problem Winston, Halsted Press, altitudes. Treatise print. Washington, New York, 1978. C. H. Sah. Hilbert's Third Problem: Scissors Congruence. Pitman, London 1979. Problem 4. Problem of the straight line as the shortest distance between two points. (Alternative geometries.) Treatise print. Problem 5. Lie's concept of a continuous group of Montgomery and Zippin. transformations without the assumption Topological Transformation Groups. of the differentiability of the functions Wiley, New York, 1955. defining the group. (Are continuous groups automatically differential Kaplansky. Lie Algebras and Locally groups?) Treatise print. Compact Groups. Chicago Univ. Press, Chicago, 1971. Problem 6. Mathematical treatment of the axioms of Leo Corry's article "Hilbert and the physics. Treatise print. Axiomatization of Physics (1894- 1905)" in the research journal Archive for History of Exact Sciences, 51 (1997). Problem 7. Irrationality and transcendence of N.I.Feldman. Hilbert's seventh certain numbers. Treatise print. problem (in Russian), Moscow state Univ, 1982, 312pp. MR 85b:11001 Problem 8. Problems of prime numbers. (The distribution of primes and the Riemann hypothesis.) Treatise print. Problem 9. Proof of the most general law of reciprocity in any number field. Treatise print. Problem 10. Determination of the solvability of a S. Chowla. The Riemann Hypothesis diophantine equation. Treatise print. and Hilbert's Tenth Problem. Gordon and Breach, New York, 1965. Maxim Vsemirnov's Hilbert's Tenth Problem page at the Steklov Institute of Yu. V. Matiyasevich. Hilbert's Tenth Mathematics at St.Petersburg. Problem. MIT Press, Cambridge, Massachusetts,1993, available on the web. Problem 11. Quadratic forms with any algebraic numerical coefficients. Treatise print. Problem 12. Extension of Kroneker's theorem on R.-P. Holzapfel. The Ball and Some abelian fields to any algebraic realm of Hilbert Problems. Springer-Verlag, rationality. Treatise print. New York, 1995. Problem 13. Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments. (Generalizes the impossibility of solving 5- th degree equations by radicals.) Treatise print. Problem 14. Proof of the finiteness of certain Masayoshi Nagata. Lectures on the complete systems of functions. Treatise fourteenth problem of Hilbert. Tata print. Institute of Fundamental Research, Bombay, 1965. Problem 15. Rigorous foundation of Schubert's enumerative calculus. Treatise print. Problem 16. Problem of the topology of algebraic Yu. Ilyashenko, and S. Yakovenko, curves and surfaces. Treatise print. editors. Concerning the Hilbert 16th problem. American Mathematical Society, Providence, R.I., 1995. B.L.J. Braaksma, G.K. Immink, and M. van der Put, editors. The Stokes Phenomenon and Hilbert's 16th Problem. World Scientific, London, 1996. Problem 17. Expression of definite forms by squares. Treatise print. Problem 18. Building up of space from congruent polyhedra. (n-dimensional crystallography groups, fundamental domains, sphere packing problem.) Treatise print. Comments on the theory of analytic functions. Problem 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic? Treatise print. Problem 20. The general problem of boundary values. (Variational problems.) Treatise non print chess. Source? Problem 21. Proof of the existence of linear differential equations having a prescribed monodromic group. Treatise non print chess. Source? Problem 22. Uniformization of analytic relations by means of automorphic functions. Treatise non print chess. Source? Problem 23. Further development of the methods of the calculus of variations. Treatise non print chess. Source? Final comments. 3.14 2 2 a = 1/2bh a = bh a2 b2 c2 = bh E = mc2