Continued fraction algorithms and Lagrange's theorem in ${\mathbb Q}_p$

TL;DR

This paper introduces new continued fraction algorithms for ${ m Q}_p$ that produce periodic expansions for quadratic elements and finite expansions for rationals, along with characterizations of purely periodic cases.

Contribution

The paper develops several novel continued fraction algorithms for ${ m Q}_p$ with complete characterizations of periodic and purely periodic expansions.

Findings

Algorithms produce eventually periodic expansions for quadratic elements.

Algorithms give finite expansions for rational numbers.

Complete characterization of purely periodic expansions.

Abstract

We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give, for each of our algorithms, the complete characterization of elements having purely periodic expansions.