Euclidean strings

Abstract

A string p=p0p1⋯pn−1 of non-negative integers is a Euclidean string if the string (p0+1)p1⋯(pn−1−1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0+p1+⋯+pn−1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stern–Brocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.