New properties of the φ-representation of integers

2025/09/19 by Shallit, Jeffrey, Vukusic, Ingrid
#FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.16150

Abstract

We prove a few new properties of the φ-representation of integers, where φ= (1+√(5))/2. In particular, we prove a 2012 conjecture of Kimberling.

Citations

• The structure of base phi expansions
• Proving Properties of φ-Representations with the Walnut Theorem-Prover
• Counting base phi representations
• On the representation of the natural numbers by powers of the golden mean
• The sum of digits function of the base phi expansion of the natural numbers
• Base phi representations and golden mean beta-expansions
• Automatic Theorem Proving in Walnut
• Representations for real numbers and their ergodic properties

Related

• Counting base phi representations
• Base phi representations and golden mean beta-expansions
• On the representation of the natural numbers by powers of the golden mean
• Shifting Zeckendorf and Chung-Graham representations
• Proving Properties of φ-Representations with the Walnut Theorem-Prover