Lagrange’s Theorem

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

Abstract

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.

Keywords

Continue Fraction Integer Solution Diophantine Equation Classical Theorem Distinct Eigenvalue
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.

References

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