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Fixed Points of the Josephus Function via Fractional Base Expansions

Yunier Bello-Cruz, Roy Quintero-Contreras

TL;DR

This work addresses the fixed-point structure of the Josephus function $J_3$ by establishing a CRT-based description of its fixed points and by encoding these points in a nonstandard base-$3/2$ numeral system. The authors derive a recurrence for the fixed-point sequence $n_p^{(\ell)}$ that depends on the divisibility parameter $\overline{m}_{\ell}$ and show that each consecutive point solves a pair of congruences via the Chinese Remainder Theorem, yielding a unique value modulo $3^{p}2^{q}$. They then construct finite, unique base-$3/2$ expansions for the fixed points using a four-step digit algorithm and prove a precise digit-recursive relation linking $n_p^{(\ell+1)}$ to $n_p^{(\ell)}$ depending on $\overline{m}_{\ell}$. This base-$3/2$ perspective generalizes the known $J_2$ binary pattern and opens questions about computing $J_3(n)$ directly from these representations and about extending the approach to $J_4$ with base $4/3$.

Abstract

In this paper, we investigate some interesting 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 allows us to derive a recursive procedure for determining the digits of their base $3/2$ expansions.

Fixed Points of the Josephus Function via Fractional Base Expansions

TL;DR

This work addresses the fixed-point structure of the Josephus function by establishing a CRT-based description of its fixed points and by encoding these points in a nonstandard base- numeral system. The authors derive a recurrence for the fixed-point sequence that depends on the divisibility parameter and show that each consecutive point solves a pair of congruences via the Chinese Remainder Theorem, yielding a unique value modulo . They then construct finite, unique base- expansions for the fixed points using a four-step digit algorithm and prove a precise digit-recursive relation linking to depending on . This base- perspective generalizes the known binary pattern and opens questions about computing directly from these representations and about extending the approach to with base .

Abstract

In this paper, we investigate some interesting properties of the fixed point sequence of the Josephus function . 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 . This result allows us to derive a recursive procedure for determining the digits of their base expansions.

Paper Structure

This paper contains 6 sections, 1 theorem, 33 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Main Results and Motivations
  3. Fixed points of $J_3$ and the Chinese Remainder Theorem
  4. Expansions in base $3/2$
  5. Characterization of $\{n_p^{(\ell)}\}_{\ell\in \mathbb{N}}$ through the base $3/2$ number system
  6. Concluding Remarks

Key Result

Theorem 1

Assume that the base $3/2$ expansion of the fixed point $n_p^{(\ell)}$ of the Josephus function $J_3$ is given by $n_p^{(\ell)} = (\hat{d_k} \cdots \hat{d_{0}})_{3/2}.$ Then, the following statements hold: If $\overline{m}_{\ell}=0$, then $n_p^{(\ell+1)}=(\hat{d_k} \cdots \hat{d_{0}} \, 1)_{3/2}$. I

Figures (1)

  • Figure 1: Josephus function $J_{3}$ for $n\le 50$.

Theorems & Definitions (4)

  • Definition : High extremal points
  • Remark
  • Theorem 1
  • proof