The abc Conjecture

  • Serge Lang


Let’s start with a theorem about polynomials. You probably think that one knows everything about polynomials. Most mathematicians would think that, including myself. It came as a surprise when R. C. Mason in 1983 discovered a new, very interesting fact about polynomials [Ma 83]. Even more remarkable, this fact actually had been already discovered by another mathematician, W. Stothers [Sto81], but people had not paid attention and I learned of Stothers’ paper only much later, from U. Zannier [Za 95] who also rediscovered some of Stothers’ results. So the history of mathematics does not always flow smoothly.


Elliptic Curf Diophantine Equation Logarithmic Derivative Irreducible Polynomial Distinct Root 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BLSTW 83]
    J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Factorization of b n ± 1, b = 2, 3, 5, 6, 7, 10, 11 up to high powers, Contemporary Math., Vol. 22, AMS, 1983Google Scholar
  2. [Da 65]
    H. Davenport, On f 3(r)-g 2(t), K. Norske Vid. Selsk. Forrh. (Trondheim) 38 (1965) pp. 86–87MathSciNetMATHGoogle Scholar
  3. [Ha 71]
    M. Hall, The diophantine equation x 3-y 2 = k, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, 1971 pp. 173–198Google Scholar
  4. [La 87]
    S. Lang, Undergraduate Algebra, Springer-Verlag, 1987, 1990. See especially Chapter IV, §9.Google Scholar
  5. [La 90]
    S. Lang, Old and new conjectured diophantine inequalities, Bull. AMS 23 (1990) pp. 37–75MATHCrossRefGoogle Scholar
  6. [Ma 83]
    R. C. Mason, Diophantine Equations over Function Fields, London Math. Soc. Lecture Note Series, Vol. 96, Cambridge University Press, 1984Google Scholar
  7. [StT 86]
    C. L. Stewart and R. Tijdeman, On the Oesterle-Masser conjecture, Monatshefte Math. 102 (1986) pp. 251–257MathSciNetMATHCrossRefGoogle Scholar
  8. [St 81]
    W. Stothers, Polynomial identities and hauptmoduln, Quart. Math. Oxford (2)32 (1981) pp. 349–370MathSciNetCrossRefGoogle Scholar
  9. [Wi 95]
    A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Annals of Math. 142 (1995) pp. 443–551MathSciNetCrossRefGoogle Scholar
  10. [Za 95]
    U. Zannier, On Davenport’s bound for the degree of f 3-g 2 and Riemann’s existence theorem, Acta Arithm. LXXI.2 (1995) pp. 107–137MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations