A Division Algorithm Approach to $p$-Adic Sylvester Expansions

TL;DR

This paper introduces a novel division algorithm-based method for constructing finite $p$-adic Sylvester expansions for rationals and extends to certain irrationals, paralleling classical algorithms.

Contribution

It presents a new $p$-adic division algorithm approach for Sylvester expansions, bridging classical and $p$-adic number representations.

Findings

Constructs finite $p$-adic Sylvester expansions for all rationals.

Extends the method to irrational $p$-adics with real embeddings.

Parallels classical Fibonacci-Sylvester algorithm.

Abstract

A method of constructing finite $p$-adic Sylvester expansions for all rationals is presented. This method parallels the classical Fibonacci-Sylvester (greedy) algorithm by iterating a $p$-adic division algorithm. The method extends to irrational $p$-adics that have an embedding in the reals.