Andrej Dujella:

Diophantine equations

Graduate course     (2006/2007)

Course description

This course will cover some of the main methods for solving diophantine equations.

We will describe in details the results and algorithms related to classical Diophantine equations, like Pellian equations and ternary quadratic forms. On these equations, the general principles for solving Diophantine equations will be illustrated: applications of results from Diophantine approximations, algebraic number theory and p-adic analysis.

We will study the modern tools from Diophantine approximations (linear forms in logarithms of algebraic numbers, hypergeometric method for rational approximations of algebraic integers), which allow us to obtain upper bounds for the size of solutions of various types of Diophantine equations. The most popular methods for the reduction of these upper bounds (Baker-Davenport method based on continued fraction, reduction using LLL-algorithm) will be described. We will illustrate by examples how the described methods lead to complete solution of various Diophantine problems. These problems will include Thue equations, integers points on elliptic curves, systems of Pellian equations and equations with recursive sequences.

It will be assumed that the students are familiar with the basic notions and results from number theory, at the level covered in the undergraduate course Introduction to Number Theory.


  1. N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, Cambridge, 1998.

  2. S. Alaca, K. S. Williams: Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 2004.

  3. W. S. Anglin: The Queen of Mathematics. An Introduction to Number Theory, Kluwer Academic Publishers, Dordrecht, 1995.

  4. H. Cohen: Number Theory. Volume I: Tools and Diophantine Equations, Springer Verlag, Berlin, 2007.

  5. H. Cohen: Number Theory. Volume II: Analytic and Modern Tools, Springer Verlag, Berlin, 2007.

  6. I. Gaal: Diophantine Equations and Power Integral Bases, Birkhauser, Boston, 2002.

  7. W. J. LeVeque: Topics in Number Theory. Volumes I and II, Dover, New York, 2002.

  8. L. J. Mordell: Diophantine Equations, Academic Press, London, 1969.

  9. T. Nagell: Introduction to Number Theory, Chelsea, New York, 1981.

  10. I. Niven, H. S. Zuckerman, H. L. Montgomery: An Introduction to the Theory of Numbers, Wiley, New York, 1991.

  11. A. Pethö: Algebraische Algorithmen, Vieweg, Braunschweig, 1999.

  12. W. M. Schmidt: Diophantine Approximation and Diophantine Equations, Springer-Verlag, Berlin, 1996.

  13. T. H. Shorey, R. Tijdeman: Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986.

  14. J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Springer-Verlag, New York, 1992.

  15. V. G. Sprindzuk: Classical Diophantine Equations, Springer, Berlin, 1993.

  16. J. Steuding: Diophantine Analysis, Chapman & Hall/CRC, Boca Raton, 2005.

  17. B. M. M. de Weger: Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.

  18. U. Zannier: Some applications of Diophantine Approximation to Diophantine Equations, Forum Editrice, Udine, 2003.

Lecture notes
(in pdf format; in Croatian)

Seminar topics

Homework exercises:

ex1   ex2   ex3   ex4   ex5   ex6

Some (useful) links

Seminar on Number Theory and Algebra (University of Zagreb)
Introduction to Number Theory - Undergraduate course (Andrej Dujella)
Cryptography - Undergraduate course (Andrej Dujella)
Elliptic curves and their applications in cryptography - Student seminar (2002/2003)
Software packages of interest to number theory
PARI/GP home page
MAGMA Calculator
Pell equation solver (Michael Zuker)
Quadratic Diophantine equation solver (Dario Alpern)
Thue equations (Clemens Heuberger)
Diophantine m-tuples page (Andrej Dujella)
Hilbert's Tenth Problem page (Maxim Vsemirnov, Yuri Matiyasevich)
Number Theory Web
Number theory groups and seminars
Recommended readings for graduate students in number theory
Henri Cohen: “Explicit methods for solving Diophantine equations”, Arizona Winter School, March 11-15, 2006
Instructional conference "Solvability of Diophantine equations", Leiden, May 7-11, 2007

Andrej Dujella home page