Length of the continued logarithm algorithm on rational inputs

TL;DR

This paper establishes an upper bound on the number of steps the continued logarithm algorithm takes to terminate on rational inputs, showing it is proportional to the logarithm of the input size, with tight bounds.

Contribution

The paper proves a tight upper bound of 2 log_2 p + O(1) steps for the continued logarithm algorithm on rational inputs, advancing understanding of its efficiency.

Findings

The algorithm terminates in at most 2 log_2 p + O(1) steps.

The bound is tight up to an additive constant.

Provides a precise complexity measure for the continued logarithm algorithm.

Abstract

The continued logarithm algorithm was introduced by Gosper around 1978, and recently studied by Borwein, Calkin, Lindstrom, and Mattingly. In this note I show that the continued logarithm algorithm terminates in at most 2 log_2 p + O(1) steps on input a rational number p/q >= 1. Furthermore, this bound is tight, up to an additive constant.