Advertisement

Diophantine Equations

  • Kenneth Ireland
  • Michael Rosen
Part of the Graduate Texts in Mathematics book series (GTM, volume 84)

Abstract

In Chapter 10 we discussed Diophantine equations over finite fields. In this chapter we consider special Diophantine equations with integral coefficients and seek integral or rational solutions. The techniques used vary from elementary congruence considerations to the use of more sophisticated results in algebraic number theory. In addition to establishing the existence or nonexistence of solutions we also obtain results of a quantitative nature, as in the determination of the number of representations of an integer as the sum of four squares. All of the equations considered in this chapter are classical, each playing an important role in the historical development of the subject.

Keywords

Rational Point Rational Solution Integral Solution Diophantine Equation Unique Factorization 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 180.
    W. J. LeVeque. A brief survey of diophantine equations. M.A.A. Studies in Mathematics, 6 (1969), 4–24.Google Scholar
  2. 39.
    G. H. Hardy. An introduction to the theory of numbers. Bull. Am. Math. Soc, 35 (1929), 778–818.zbMATHCrossRefGoogle Scholar
  3. 146.
    T. L. Heath. Diophantus of Alexandria: A Study in the History of Greek Algebra, New York: Dover, 1964.zbMATHGoogle Scholar
  4. 152.
    J. E. Hofmann. Über Zahlentheoretische Methoden Fermats und Eulers, ihre Zusammenhänge und ihre Bedeutung. Arch. Hist. Exact Sci (1960–62), 122–159.Google Scholar
  5. 84.
    W. W. Adams and L. J. Goldstein. Introduction to Number Theory. Englewood Cliffs, N.J.: Prentice-Hall, 1976.zbMATHGoogle Scholar
  6. 40.
    G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. 4th ed., New York: Oxford University Press, 1960.zbMATHGoogle Scholar
  7. 230.
    J. V. Uspensky and M. A. Heaslet, New York: McGraw-Hill, 1939.Google Scholar
  8. 22.
    H. Davenport. The Higher Arithmetic. London: Hutchinson, 1968.Google Scholar
  9. 61.
    I. Niven and H. S. Zuckerman. An Introduction to the Theory of Numbers. 2nd ed. New York: Wiley, 1966.zbMATHGoogle Scholar
  10. 235.
    A. Weil. Two lectures on number theory: Past and present. L’Enseignement Math, XX (1973), 81–110. Also in: A.’ Weil, Oeuvres Scientifiques, Vol. III, pp. 279–302. New York: Springer-Verlag, 1979.Google Scholar
  11. 189.
    L. J. Mordell. Diophantine Equations. New York: Academic Press, 1969.zbMATHGoogle Scholar
  12. 170.
    S. Lang. Diophantine Geometry. New York: Wiley-Interscience, 1962.zbMATHGoogle Scholar
  13. 190.
    L. J. Mordell. Review of S. Lang’s diophantine geometry. Bull. Am. Math. Soc, 70 (1964), 491–498.MathSciNetCrossRefGoogle Scholar
  14. 172.
    S. Lang. Review of L. J. Mordell’s diophantine equations. Bull. Am. Math. Soc, 76 (1970), 1230–1234.CrossRefGoogle Scholar
  15. 53.
    S. Lang. Some theorems and conjectures on diophantine equations. Bull. Am. Math. Soc, 66 (1960), 240–249.zbMATHCrossRefGoogle Scholar
  16. 173.
    S. Lang. Higher dimensional diophantine problems. Bull. Am. Math. Soc., 80, no. 5 (1974), 779–787.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  • Kenneth Ireland
    • 1
  • Michael Rosen
    • 2
  1. 1.Department of MathematicsUniversity of New BrunswickFrederictonCanada
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

Personalised recommendations