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New properties of the $\varphi$-representation of integers

Jeffrey Shallit, Ingrid Vukusic

Abstract

We prove a few new properties of the $\varphi$-representation of integers, where $\varphi = (1+\sqrt{5})/2$. In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we used the Walnut theorem-prover, and in one proof, ChatGPT 5.

New properties of the $\varphi$-representation of integers

Abstract

We prove a few new properties of the -representation of integers, where . In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we used the Walnut theorem-prover, and in one proof, ChatGPT 5.

Paper Structure

This paper contains 3 sections, 8 theorems, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Shevelev's set and Kimberling's conjecture
  3. Other results

Key Result

Lemma 2

Suppose the $\varphi$-expansion of a non-negative real number $x$ is antipalindromic. Then $x$ is an integer if and only if all the exponents appearing in its expansion are even.

Figures (4)

  • Figure 1: Automaton accepting $(x,y)$ such that $x.y^R$ is the $\varphi$-representation of some integer $n \geq 0$.
  • Figure 2: Automaton accepting the Zeckendorf representations of the members of sequence https://oeis.org/A178482.
  • Figure 3: Automaton accepting Zeckendorf representation of those $n$ whose $\varphi$-representation has exactly one even exponent.
  • Figure 4: Automaton accepting Zeckendorf representation of those $n$ whose $\varphi$-representation has exactly one odd exponent.

Theorems & Definitions (21)

  • Example 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Example 6
  • Proposition 7
  • ...and 11 more