P -adic Continued Fractions

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a periodic continued fraction for √ m. In Jerzy Browkin’s 1978 and 2000 papers, Continued Fractions in Local Fields, I and II, respectively, Browkin presents two definitions for a p-adic continued fraction and presents several algorithms for computing continued fraction approximations to p-adic square roots with the end-goal of finding periodic continued fraction expansions. This paper will serve as an introduction to p-adic numbers and as an exploration of the definitions and algorithms associated with p-adic continued fractions. 1 Definitions Definition: Cauchy Sequence: Let {xn}n∈N be a sequence. Then, {xn}n∈N is a Cauchy Sequence if for all > 0 there exists N ∈ N such that, for all n ≥ N , |xn − xn+1| < . Note that the summation of a Cauchy Sequence, ∑∞ i=0 xi, converges. Definition: Valuation: Let K be a field. A valuation on K is a function | · | : K −→ R with the following properties: 1. |a| ≥ 0 for all a ∈ K, and |x| = 0 if and only if x = 0. 2. |a ∗ b| = |a| · |b| for all a, b ∈ K.