Necessary and sufficient conditions for convergence of integer continued fractions

TL;DR

This paper introduces a simple test to determine the convergence of integer continued fractions, including whether their limits are rational or irrational, and provides a geometric interpretation using Farey graphs.

Contribution

It presents a novel, straightforward convergence test for integer continued fractions and links it to a geometric visualization via Farey graphs, enhancing understanding.

Findings

A simple convergence test for integer continued fractions.

The test distinguishes rational and irrational limits.

Geometric interpretation using Farey graphs offers deeper insight.

Abstract

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.