Geometric Methods in Number Theory

  • Paul Erdős
  • János Surányi
Part of the Undergraduate Texts in Mathematics book series (UTM)


In the proofs of the previous chapters we often used geometrical considerations. We will present one more such proof. Wilson’s theorem (Theorem 2.14) was easily proved in two different ways using congruences. We will now give a proof of it that does not use congruences.


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  1. 3.
    See K. Härtig, J. Surányi: Periodica Math. Hung. 6 (1975), pp. 235–240.MATHCrossRefGoogle Scholar
  2. 4.
    In addition to his articles, he wrote two books that thoroughly deal with this subject: Geometrie der Zahlen (Leipzig, 1897) and Diophantische Approximationen (Leipzig, 1907). A modern treatment is given in J.W. S. Cassels: An Introduction to the Geometry of Numbers (Springer, 1959), and further in the encyclopedic work P. Gruber, C.G. Lekkerkerker: Geometry of numbers (North Holland, 1987).Google Scholar
  3. 5.
    See Mat és Fiz. Lapok 50. (1943), pp. 182–183, problem 12 (in Hungarian).Google Scholar
  4. 6.
    G. Pick: Lotos Prag. (2) 19 (1900), pp. 311–319.MathSciNetGoogle Scholar
  5. 9.
    G. L. Alexanderson, J. Pedersen: The Oregon Math. Teacher, 1985. The presented format of the proof was given by Árpád Somogyi, who found it independently of Pólya.Google Scholar
  6. 10.
    This problem appeared on the Miklós Schweizer Memorial Competition in 1964. This is a national mathematics competition in Hungary for high school students. See Matematikai Lapok 16 (1965), pp. 92–113, Problem 4 (in Hungarian) and Contests in Higher Mathematics, G.J. Székely (ed.), Springer, 1996, pp. 5 and 247–249.Google Scholar
  7. 11.
    G. Blrkhoff (1914), referred to in H. F. Blichfeldt: Transactions Amer. Math. Soc. 15 (1914), pp. 227–235. W. Scherrer: Math. Annalen 89 (1923), pp. 255–259, and Dissertation, Universität Zürich, 1923. Gy. Hajós: Acta Sei. Math. Szeged, 6 (1934), pp. 224–225.Google Scholar
  8. 12.
    E. Erhart: Comptes Rendus Acad. Sei. Paris 240 (1955), pp. 483–485.Google Scholar
  9. 13.
    This proof was supplied by P. Turán, who adapted an idea of Hermite, who proved a related theorem about the sum of four squares (as we will see in Theorem 7.4′); See H. Davenport: Math. Gazette 31 (1947), pp. 206–210.MathSciNetMATHCrossRefGoogle Scholar
  10. 14.
    K. Mahler: Quarterly Journal of Math., Oxford Ser. 17 (1946), pp. 16–18.MathSciNetMATHCrossRefGoogle Scholar
  11. 15.
    H. Davenport, K. Mahler: Duke Math. Journ. 13 (1946), pp. 105–111.MathSciNetMATHCrossRefGoogle Scholar
  12. 16.
    See T. W. Chaundy: Quarterly Journal of Mathematics, Oxford Ser. 17(67) (1946), pp. 166–192. The statement of Theorem 13 there is due to Gauss.MathSciNetMATHCrossRefGoogle Scholar
  13. 17.
    A. Korkine, G. Zolotareff: Math. Annalen 6 (1873), pp. 366–389.MathSciNetMATHCrossRefGoogle Scholar
  14. 18.
    H. Davenport: Proc. London Math. Soc. 44 (1938), pp. 412–431, and Journal London Math. Soc. 16 (1941), pp. 98–101. For further results and problems see the books of J. W. S. Cassels, and of P. M. Gruber and C. G. Lekkerkerker mentioned in footnote 4.MathSciNetMATHCrossRefGoogle Scholar
  15. 20.
    See A. A. Markoff: Math. Annalen 15 (1879), pp. 381–406. For a different proof, further references, and problems, see the book of J. W. S. Cassels referred to in footnote 4, pp. 18–44.MATHCrossRefGoogle Scholar
  16. 21.
    B. N. Delone: Izvesztyija Akad. Nauk SSSR. Ser. Mat. 11 (1947), pp. 505–538 (in Russian). In German, see Sowjetwissenschaft 2 (1948), pp. 178–210. Cf. also J. Surányi: Acta Sei. Math. Szeged 22 (1961), pp. 85–90.MathSciNetMATHGoogle Scholar
  17. 22.
    D. B. Sawyer: Journal London Math. Soc. 23 (1948), pp. 250–251; E. S. Barnes, H.P.F. Swinnerton-Dyer: Acta Mathematica 87 (1952), pp. 259–323; ibid. 88 (1952), pp. 279–316; and ibid. 92 (1954), pp. 199–234. B. J. Birch: Proc. Cambridge Phil. Soc. 53 (1956), p. 536.Google Scholar

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© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Paul Erdős
  • János Surányi
    • 1
  1. 1.Department of Algebra and Number TheoryEõtvõs Loránd UniversityBudapestHungary

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