Lagrange’s Theorem

  • Oleg Karpenkov
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)


The aim of this chapter is to study questions related to the periodicity of geometric and regular continued fractions. The main object here is to prove Lagrange’s theorem stating that every quadratic irrationality has a periodic continued fraction, conversely that every periodic continued fraction is a quadratic irrationality. One of the ingredients to the proof of Lagrange theorem is the classical theorem on integer solutions of Pell’s equation
$$m^2-dn^2=1. $$
So, there is a strong relation between periodic fractions and quadratic irrationalities.

We start with the study of so-called Dirichlet groups, which are the subgroups of \(\operatorname{GL}(2,\mathbb{Z})\) preserving certain pairs of lines. These groups are closely related to the periodicity of sails. The structure of a Dirichlet group is induced by the structure of the group of units in orders (we will discuss this later in more detail for the multidimensional case; here we restrict ourselves to the simplest two-dimensional case). We also show how to take nth roots of two-dimensional matrices using Gauss’s reduction theory. Finally we study the solutions of Pell’s equation and prove Lagrange’s theorem.


Continue Fraction Integer Solution Diophantine Equation Classical Theorem Distinct Eigenvalue 
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  1. 129.
    J.-L. Lagrange, Solution d’un problème d’arithmétique, in Œuvres de Lagrange, vol. 1 (Gauthier-Villars, Paris, 1867–1892), pp. 671–732 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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