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Integers Having $F_{2k}$ in Both Zeckendorf and Chung-Graham Decompositions

Lucas Bustos, Hung Viet Chu, Minchae Kim, Uihyeon Lee, Shreya Shankar, Garrett Tresch

TL;DR

This work provides a complete characterization of positive integers for which the term $F_{2k}$ appears in both Zeckendorf and Chung-Graham decompositions. It introduces and exploits the golden string $\mathcal{S}$ as a unifying combinatorial encoding that links the two decompositions, through precise table constructions for $A_{2k}$ and $B_{2k}$ and their recursive expansions. The main result yields an explicit parametric description $\{nF_{2k}+\lfloor(n-1)\phi\rfloor F_{2k+1}+j : n\in\mathbb{N}, 0\le j\le F_{2k-1}-1\}$ for the integers with $F_{2k}$ in both decompositions, and an analogous characterization on the $Z(n)\cap CG(n)$ side facilitated by the correspondence with letters in $\mathcal{S}$. Together, these findings deepen understanding of the interaction between Zeckendorf and Chung-Graham representations and reveal a deep, order-preserving structure governed by the golden ratio.

Abstract

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed Fibonacci numbers with coefficients $0,1$, or $2$ such that between two coefficients $2$, there is a coefficient $0$. For each $k\ge 1$, we find the set of all positive integers having $F_{2k}$ in both of their Zeckendorf and Chung-Graham decompositions.

Integers Having $F_{2k}$ in Both Zeckendorf and Chung-Graham Decompositions

TL;DR

This work provides a complete characterization of positive integers for which the term appears in both Zeckendorf and Chung-Graham decompositions. It introduces and exploits the golden string as a unifying combinatorial encoding that links the two decompositions, through precise table constructions for and and their recursive expansions. The main result yields an explicit parametric description for the integers with in both decompositions, and an analogous characterization on the side facilitated by the correspondence with letters in . Together, these findings deepen understanding of the interaction between Zeckendorf and Chung-Graham representations and reveal a deep, order-preserving structure governed by the golden ratio.

Abstract

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed Fibonacci numbers with coefficients , or such that between two coefficients , there is a coefficient . For each , we find the set of all positive integers having in both of their Zeckendorf and Chung-Graham decompositions.
Paper Structure (8 sections, 9 theorems, 36 equations, 4 tables)

This paper contains 8 sections, 9 theorems, 36 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The golden string $\mathcal{S}$
  3. The Zeckendorf and Chung-Graham tables
  4. The Zeckendorf table
  5. The Chung-Graham table
  6. Integers $n$ in $A_{2k}$ with $F_{2k}$ or $2F_{2k}$ in $CG(n)$
  7. Integers $n$ with $F_{2k}\in Z(n)\cap CG(n)$
  8. Integers $n$ in $B_{2k}$ with $F_{2k}$ in $Z(n)$

Key Result

Theorem 1.1

CG Every positive integer $n$ can be uniquely represented as a sum so that if $c_{i_1} = c_{i_2} = 2$, then there is a $j$ between $i_1$ and $i_2$ such that $c_j = 0$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 3 more