Roots with common tails

TL;DR

This paper characterizes cubic polynomials with three real roots sharing the same continued fraction tail and explores similar phenomena in lower-degree polynomials.

Contribution

It provides a complete characterization of cubic polynomials with roots having common continued fraction tails and extends the analysis to lower-degree cases.

Findings

Identifies conditions for roots to share a common continued fraction tail

Characterizes all such cubic polynomials over the integers

Examines similar properties in polynomials of degree less than three

Abstract

Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other polynomials of low degree.