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Hilbert, Mark

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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