A new algorithm for p-adic continued fractions

TL;DR

This paper introduces an improved $p$-adic continued fraction algorithm that demonstrates enhanced periodicity properties and produces more periodic expansions for quadratic irrationals, moving closer to an analogue of Lagrange's theorem.

Contribution

The paper presents a modified algorithm for $p$-adic continued fractions with better periodicity properties and increased effectiveness in generating periodic expansions for quadratic irrationals.

Findings

The new algorithm shows improved periodicity properties.

Periodic expansions have a pre-period of one for square roots of integers.

It produces more periodic continued fractions than Browkin's algorithm.

Abstract

Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still not known if a $p$--adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkin's algorithms. This algorithm is considered one of the best at the present time. Our new algorithm shows better properties of periodicity. We show for the square root of integers that if our algorithm produces a periodic expansion, then this periodic expansion will have pre-period one. It appears experimentally that our algorithm produces more periodic continued fractions for quadratic irrationals than Browkin's algorithm. Hence, it is closer to an algorithm to which an analogue of Lagrange's Theorem would apply.