Famous Problems in Mathematics

  • Israel Kleiner


The above is the title of a one-semester course at the third-year level offered in the department of mathematics at my University. The course has a significant historical component, but it is not a course in the history of mathematics. The historical perspective is, however, essential. One of the objectives of the course is to make students aware that mathematics has a history, and that it may be interesting, useful, and important to bring history to bear on the study of mathematics. (Some technical details about the course are given at the end of the chapter.)


Integral Domain Diophantine Equation Prime Number Theorem Symbolical Algebra Transcendental Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A. A. A1bert (ed.), Studies in Modern Algebra, Math. Assoc. of Amer., 1963.Google Scholar
  2. 2.
    A. D. A1eksandrov et al., Mathematics: Its Content, Methods, and Meaning, 3 vols., M.I.T. Press, 1963.Google Scholar
  3. 3.
    R. B. J. T. Allenby, Rings, Fields and Groups, Edward Arnold Publ., 1983.Google Scholar
  4. 4.
    T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.Google Scholar
  5. 5.
    S. F. Barker, Philosophy of Mathematics, Prentice-Hall, 1964.Google Scholar
  6. 6.
    E. T. Bell, Men of Mathematics, Simon & Schuster, 1965.Google Scholar
  7. 7.
    E. T. Bell, The Development of Mathematics, 2nd ed., McGraw-Hill, 1945.Google Scholar
  8. 8.
    P. Benacerraf & H. Putnam, Philosophy of Mathematics: Selected Readings, Prentice-Hall, 1964 (reprinted by Cambridge Univ. Press, 1983).Google Scholar
  9. 9.
    N. L Biggs, E. K. Lloyd, and R. J. Wilson, Graph Theory: 1736–1936, Oxford Univ. Press, 1986.Google Scholar
  10. 10.
    G. Birkhoff and S. Mac Lane, A Survey of Modern Algebra, 4th ed., Macmillan, 1977 (orig. 1941).Google Scholar
  11. 11.
    E. D. Bolker, Elementary Number Theory: An Algebraic Approach, W.A. Benjamin, 1970.Google Scholar
  12. 12.
    R. Bonola, Non-Euclidean Geometry, Dover, 1955 (orig. 1911).Google Scholar
  13. 13.
    Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, 1966.Google Scholar
  14. 14.
    U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer, 1986.Google Scholar
  15. 15.
    C. B. Boyer, A History of Mathematics, 2nd ed., revised by U. Merzbach, Wiley, 1989.Google Scholar
  16. 16.
    F. E. Browder, Does pure mathematics have a relation to the sciences? Amer. Scientist 64 (Sept./Oct.1976) 542–549.Google Scholar
  17. 17.
    D. M. Burton, The History of Mathematics: An Introduction, 6th ed., McGraw-Hill, 2007.Google Scholar
  18. 18.
    F. Cajori, History of exponential and logarithmic concepts, Amer. Math. Monthly 20 (1913) 5–14, 35–47, 75–84, 107–117, 148–151, 173–182, and 205–210.Google Scholar
  19. 19.
    D. M. Campbell and J. C. Higgins (eds.), Mathematics: People, Problems, Results, 3 Vols., Wadsworth, 1984.Google Scholar
  20. 20.
    R. Courant and H. Robbins, What is Mathematics? Oxford Univ. Press, 1941.Google Scholar
  21. 21.
    J. N. Crossley, The Emergence of Number, World Scientific,1987.Google Scholar
  22. 22.
    J. N. Crossley et al., What is Mathematical Logic? Oxford Univ. Press, 1972.Google Scholar
  23. 23.
    T. Dantzig, Number: The Language of Science, 4th ed., Free Press, 1967 (orig. 1930).Google Scholar
  24. 24.
    J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard Univ. Press, 1979MATHGoogle Scholar
  25. 25.
    P. J. Davis and R. Hersh, The Mathematical Experience, Birkhäuser, 1981.Google Scholar
  26. 26.
    H. De Long, A Profile of Mathematical Logic, Addison Wesley, 1970.Google Scholar
  27. 27.
    R. A. De Millo et al, Social processes and proofs of theorems and programs, Math. Intelligencer 3 (1980) 31– 40.CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    L. E. Dickson, Fermat’s Last Theorem and the origin and nature of the theory of algebraic numbers, Ann. Math. 18 (1916–17) 161–187.Google Scholar
  29. 29.
    J. Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.Google Scholar
  30. 30.
    H. Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution, Dover, 1965.Google Scholar
  31. 31.
    U. Dudley, Formulas for primes, Math. Mag. 56 (1983) 17–22.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    W. Dunham, Journey Through Genius: The Great Theorems of Mathematics, Wiley, 1990.Google Scholar
  33. 33.
    H.-D. Ebbinghaus et al, Numbers, Springer, 1990.Google Scholar
  34. 34.
    H. M. Edwards, Fermat’s Last Theorem, Scientific Amer. 239 (Oct. 1978) 104–122.CrossRefGoogle Scholar
  35. 35.
    H. M. Edwards, Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, Springer, 1977.Google Scholar
  36. 36.
    H. M. Edwards, Riemann’s Zeta Function, Academic Press, 1974Google Scholar
  37. 37.
    L. Euler, Discovery of a most extraordinary law of numbers concerning the sum of their divisors, Opera Omnia, Ser. 1, Vol. 2, pp. 241–253. (E175 in the Eneström index.)Google Scholar
  38. 38.
    H. Eves. An Introduction to the History of Mathematics, 6th ed., Saunders College Publ., 1990 (orig. 1953).Google Scholar
  39. 39.
    H. Eves, Great Moments in Mathematics (after 1650), Math. Assoc. of Amer., 1981.Google Scholar
  40. 40.
    H. Eves, Great Moments in Mathematics (before 1650), Math. Assoc. of Amer., 1980.Google Scholar
  41. 41.
    H. Eves and C. Newsom, An Introduction to the Foundations and Fundamental Concepts of Mathematics, Holt, Rinehart & Winston, 1958.Google Scholar
  42. 42.
    S. Feferman, What does logic have to tell us about mathematical proofs? Math. Intelligencer 2 (1979) 20–24.CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    D. Fendel, Prime producing polynomials and principal ideal domains, Math. Mag. 58 (1985) 204–210.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    H. Flanders, A tale of two squares – and two rings, Math. Mag. 58 (1985) 3–11.CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    A. Gardiner, Infinite Processes: Background to Analysis, Springer, 1982.Google Scholar
  46. 46.
    M. Gardner, Patterns in primes are a clue to the strong law of small numbers, Scientific Amer. 243 (Dec. 1980) 18–28.Google Scholar
  47. 47.
    C. C. Gillispie (ed.), Dictionaryof Scientific Biography, 16vols., Scribner’s, 1970–1980.Google Scholar
  48. 48.
    J. R. Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, A K Peters, 1998.Google Scholar
  49. 49.
    J. Grabiner, Changing attitudes toward mathematical rigor: Lagrange and analysis in the eighteenth and nineteenth centuries. In Epistemological and Social Problems of the Sciences in the Early 19th Century, H. Jahnke & M. Otte (eds.), D. Reidel, 1981, pp. 311–330.Google Scholar
  50. 50.
    J. Grabiner, Is mathematical truth time-dependent? Amer. Math. Monthly 81 (1974) 354–365.CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    I. Grattan-Guinness (ed.), From the Calculus to Set Theory, 1630- 1910: An Introductory History, Princeton Univ. Press, 2000.Google Scholar
  52. 52.
    I. Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, MIT Press, 1970.Google Scholar
  53. 53.
    J. Gray, Worlds out of Nothing: A Course in the History of Geometry in the 19 th Century, Springer, 2007.Google Scholar
  54. 54.
    M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 2nd ed., W. H. Freeman & Co., 1980.Google Scholar
  55. 55.
    E. Grosswald, Topics from the Theory of Numbers, 2nd ed., Birkhäuser, 1984.Google Scholar
  56. 56.
    R. K. Guy, Conway’s prime producing machine, Math. Mag. 56 (1983) 2–33.CrossRefMathSciNetGoogle Scholar
  57. 57.
    R. W. Hamming, The unreasonable effectiveness of mathematics, Amer. Math. Monthly 87 (1980) 81–90.CrossRefMathSciNetGoogle Scholar
  58. 58.
    G. Hanna, Rigorous Proof in Mathematics Education, Ontario Institute for Studies in Educ. Press, 1983.Google Scholar
  59. 59.
    G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1960 (orig. 1938).MATHGoogle Scholar
  60. 60.
    T. Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development, Chelsea, 1970.Google Scholar
  61. 61.
    R. Hersh, What is Mathematics, Really?, Oxford Univ. Press, 1997.Google Scholar
  62. 62.
    I. N. Herstein, Topics in Algebra, Blaisdell Publ. Co., 1964.Google Scholar
  63. 63.
    D. R. Hofstadter, Analogies and metaphors to explain Gödel’s Theorem, Two Yr. Coll. Math. Jour. 13 (1982) 98–114.CrossRefGoogle Scholar
  64. 64.
    J. P. Jones et al., Diophantine representation of the set of primes, Amer. Math. Monthly 83 (1976) 449–464.CrossRefMATHMathSciNetGoogle Scholar
  65. 65.
    M. Kac & S. M. Ulam, Mathematics and Logic: Retrospect and Prospects, The New Amer. Library, 1969.Google Scholar
  66. 66.
    I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Springer-Verlag, 1989.Google Scholar
  67. 67.
    E. Kasner & J. R. Newman, Mathematics and the Imagination, Simon & Schuster, 1967 (orig. 1940).Google Scholar
  68. 68.
    J. G. Kemeny, Rigor vs. intuition in mathematics, Math. Teacher 54 (1961) 66–74.Google Scholar
  69. 69.
    I. Kleiner, A History of Abstract Algebra, Birkhäuser, 2007.Google Scholar
  70. 70.
    I. Kleiner & S. Avital, The relativity of mathematics, Math. Teacher 77 (1984) 554–558, 562.Google Scholar
  71. 71.
    M. Kline, Mathematics: The Loss of Certainty, Oxford Univ. Press, 1980.Google Scholar
  72. 72.
    M. Kline (ed.), Mathematics: An introduction to its Spirit and Use, W.H. Freeman & Co., 1979.Google Scholar
  73. 73.
    M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford Univ. Press, 1972.MATHGoogle Scholar
  74. 74.
    M. Kline (ed.), Mathematics in the Modern World, W. H. Freeman & Co., 1968.Google Scholar
  75. 75.
    M. Kline, Mathematics in Western Culture, Oxford Univ. Press, 1953.MATHGoogle Scholar
  76. 76.
    G. Kolata, Does Gödel’s Theorem matter for mathematics? Science 218 (Nov. 1982) 779–780.CrossRefMATHMathSciNetGoogle Scholar
  77. 77.
    G. Kolata, Mathematical proofs: the genesis of reasonable doubt, Science 192 (June 1976) 989–990.CrossRefGoogle Scholar
  78. 78.
    A.G. Kurosh, Lectures on General Algebra, Chelsea Publ. Co., 1963.Google Scholar
  79. 79.
    I. Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge Univ. Press, 1976.MATHGoogle Scholar
  80. 80.
    R. E. Langer, Fourier series: the genesis and evolution of a theory, Amer. Math. Monthly 54 Supplement (1947) 1–86.Google Scholar
  81. 81.
    R. Laugenbacher and D. Pengelley, Mathematical Expeditions: Chronicles by the Explorers, Springer, 1999.Google Scholar
  82. 82.
    D. Laugwitz, Controversies about numbers and functions. In The Growth of Mathematical Knowledge, E. Grosholz and H. Berger (eds), Kluwer, 2000, pp. 177–198.Google Scholar
  83. 83.
    Leapfrogs, Imaginary Logarithms, Leapfrogs, 1978.Google Scholar
  84. 84.
    J. R. Manheim, The Genesis of Point Set Topology, Pergamon Press, 1964.Google Scholar
  85. 85.
    Yu. I. Manin, How convincing is a proof? Math. Intelligencer 2 (1979) 17–18.CrossRefMathSciNetGoogle Scholar
  86. 86.
    P. Marchi, The controversy between Leibniz and Bernoulli on the nature of the logarithms of negative numbers. In Akten das II Inter. Leibniz-Kongress, Bnd II, 1974, pp. 67–75.Google Scholar
  87. 87.
    K. O. May, The Impossibility of a division algebra of vectors in three dimensional space, Amer. Math. Monthly 73 (1966) 289–291.CrossRefMathSciNetGoogle Scholar
  88. 88.
    N. H. McCoy, Introduction to Modern Algebra, Allyn & Bacon, 1968.Google Scholar
  89. 89.
    R. K. Merton, Singletons and multiples in scientific discovery: a chapter in the sociology of science, Proc. Amer. Philos. Soc. 105 (1961) 470–486.Google Scholar
  90. 90.
    H. Meschkowski, Evolution of Mathematical Thought, Holden-Day, 1965.Google Scholar
  91. 91.
    H. Meschkowski, Noneuclidean Geometry, Academic Press, 1964.Google Scholar
  92. 92.
    H. Meschkowski, Ways of Thought of Great Mathematicians, Holden-Day, 1964.Google Scholar
  93. 93.
    H. L. Montgomery, Zeta zeros on the critical line, Amer. Math. Monthly 86 (1979) 43–45.CrossRefMATHMathSciNetGoogle Scholar
  94. 94.
    E. Nagel and J. R. Newman, Gödel’s Proof, New York Univ. Press, 1958.MATHGoogle Scholar
  95. 95.
    J. R. Newman (ed.), The World of Mathematics, 4 vols., Simon &. Schuster, 1956.Google Scholar
  96. 96.
    I. Niven, Numbers: Rational and Irrational, Random House, 1961.Google Scholar
  97. 97.
    C. S. Ogilvy and. J. T. Anderson, Excursions in Number Theory, Oxford Univ. Press, 1966.Google Scholar
  98. 98.
    O. Ore, Number Theory and Its History, McGraw-Hill, 1948.Google Scholar
  99. 99.
    J. Pierpont, Mathematical rigor, past and present, Bull. Amer. Math. Soc. 34 (1928) 23–53.CrossRefMATHMathSciNetGoogle Scholar
  100. 100.
    H. Pollard and. H. G. Diamond, The Theory of Algebraic Numbers, 2nd ed., Math. Assoc. of Amer., 1975.Google Scholar
  101. 101.
    C. Pomerance, The search for primes, Scientific Amer. 247 (Dec. 1982) 136–147.CrossRefGoogle Scholar
  102. 102.
    C. Pomerance, Recent developments in primality testing, Math. Intelligencer 3 (1981) 97–105.CrossRefMATHMathSciNetGoogle Scholar
  103. 103.
    H. Pycior, Augustus De Morgan’s algebraic work: the three stages, Isis 74 (1983) 211–226.CrossRefMATHMathSciNetGoogle Scholar
  104. 104.
    H. Pycior, Historical roots of confusion among beginning algebra students: a newly discovered manuscript, Math. Mag. 55 (1982) 150–156.CrossRefMATHMathSciNetGoogle Scholar
  105. 105.
    H. Pycior, George Peacock and the British origins of symbolical algebra, Hist. Math. 8 (1981) 23–45.CrossRefMathSciNetGoogle Scholar
  106. 106.
    F. Richman, Number Theory: An Introduction to Algebra, Brooks/Cole Publ. Co., 1971.Google Scholar
  107. 107.
    R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite, Birkhäuser, 1982.Google Scholar
  108. 108.
    W. L. Schaaf (ed.), Our Mathematical Heritage, Macmillan, 1963Google Scholar
  109. 109.
    O. Shisha, Mathematically civilized, Notices Amer. Math. Soc. 30 (1983) 603.Google Scholar
  110. 110.
    G. F. Simmons, Differential Equations, with Applications and Historical Notes, McGraw-Hill, 1972.Google Scholar
  111. 111.
    C. Small, A simple proof of the Four-Square theorem, Amer. Math. Monthly 89 (1982) 59–61.CrossRefMATHMathSciNetGoogle Scholar
  112. 112.
    G. C. Smith, De Morgan and the laws of Algebra, Centaurus 25 (1981) 50–70.CrossRefMATHMathSciNetGoogle Scholar
  113. 113.
    R. M. Smullyan, What is the Name of this Book? The Riddle of Dracula and Other Logical Puzzles, Prentice-Hall, 1978.Google Scholar
  114. 114.
    E. Sondheimer and A. Rogerson, Numbers and Infinity: A Historical Account of Mathematical Concepts, Cambridge Univ. Press, 1981.MATHGoogle Scholar
  115. 115.
    H. Stark, An Introduction to Number Theory, Markham Publ. Co., 1970.Google Scholar
  116. 116.
    L. A. Steen (ed.), Mathematics Today: Twelve Informal Essays, Springer, 1978.Google Scholar
  117. 117.
    I. Stewart, The science of significant form, Math. Intelligencer 3 (1981) 50–58.CrossRefGoogle Scholar
  118. 118.
    J. Stillwell, The Four Pillars of Geometry, Springer, 2005.Google Scholar
  119. 119.
    H. Tietze, Famous Problems of Mathematics, Graylock Press, 1965.Google Scholar
  120. 120.
    V. M. Tikhomirov, Stories About Maxima and Minima, Amer. Math. Society, 1990.MATHGoogle Scholar
  121. 121.
    R. J. Trudeau, Introduction to Graph Theory, Dover, 1993.Google Scholar
  122. 122.
    B. L. Van der Waerden, Hamilton’s discovery of quaternions, Math. Mag. 49 (1976) 227–234.CrossRefMATHMathSciNetGoogle Scholar
  123. 123.
    E. B. Van Vleck, The influence of Fourier Series upon the development of mathematics, Science 39 (1914) 113–124.CrossRefGoogle Scholar
  124. 124.
    N. Ya. Vilenkin, Stories About Sets, Academic Press, 1968. (Translated by A. Shenitzer.)Google Scholar
  125. 125.
    A. Weil, Number Theory: An approach Through History, Birkhäuser, 1984.Google Scholar
  126. 126.
    H. Weyl, A half-century of mathematics, Amer. Math. Monthly 58 (1951) 523–553.CrossRefMATHMathSciNetGoogle Scholar
  127. 127.
    E. P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure &. Appl. Math. 13 (1960) 1–14.Google Scholar
  128. 128.
    R. L. Wilder, Mathematics As a Cultural System, Pergamon Press, 1981.Google Scholar
  129. 129.
    R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study, Wiley, 1968.Google Scholar
  130. 130.
    R. L. Wilder, Relativity of standards of mathematical rigor, Dict. of the Hist. of Ideas 3 (1968) 170–177.Google Scholar
  131. 131.
    R. L. Wilder, The role of intuition, Science 156 (1967) 605–610.CrossRefGoogle Scholar
  132. 132.
    R. L. Wilder, The role of the axiomatic method, Amer. Math. Monthly 74 (1967) 115–127.CrossRefMATHMathSciNetGoogle Scholar
  133. 133.
    R. L. Wilder, Introduction to the Foundations of Mathematics, Wiley, 1965.Google Scholar
  134. 134.
    R. L. Wilder, The origin and growth of mathematical concepts, Bull. Amer. Math. Soc. 59 (1953) 423–448.CrossRefMATHMathSciNetGoogle Scholar
  135. 135.
    R. L. Wilder, The nature of mathematical proof, Amer. Math. Monthly 51 (1944) 309–323.CrossRefMathSciNetGoogle Scholar
  136. 136.
    R. Wilson, Four Colours Suffice: How the Map Problem was Solved, Penguin, 2002.Google Scholar
  137. 137.
    D. Zagier, The first 50 million prime numbers, Math. Intelligencer 0 (1977) 7–19.Google Scholar
  138. 138.
    L. Zippin, Uses of Infinity, Random House, 1962.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations