Fixed Points of the Josephus Function via Fractional Base Expansions

TL;DR

This paper explores the fixed points of the Josephus function J_3, revealing their structure through fractional base expansions and establishing a link with the Chinese Remainder Theorem, leading to a recursive digit determination method.

Contribution

It introduces a novel connection between Josephus fixed points and fractional base expansions, providing a recursive approach to their digit calculation.

Findings

Identified a numerical pattern in fixed point digits in base 3/2.

Established a link between fixed points and the Chinese Remainder Theorem.

Developed a recursive procedure for digit determination in fractional base expansions.

Abstract

In this paper, we investigate properties of the fixed point sequence of the Josephus function $J_3$. First, we establish a connection between this sequence and the Chinese Remainder Theorem. Next, we identify a clear numerical pattern for the digits of two consecutive fixed points when they are written in a non-standard fractional number system in base $3/2$. This result enables us to derive a recursive procedure for determining the digits of their base $3/2$ expansions.