Zeckendorf representations with at most two terms to x-coordinates of Pell equations

TL;DR

This paper characterizes positive squarefree integers for which the Pell equation has multiple solutions with X-coordinates having simple Zeckendorf representations, advancing understanding of number representations in Pell solutions.

Contribution

It identifies all squarefree integers where Pell solutions' X-coordinates have at most two-term Zeckendorf representations, a novel classification in number theory.

Findings

Identified all such squarefree integers d

Established conditions for multiple solutions with simple Zeckendorf representations

Enhanced understanding of Pell solutions and number representations

Abstract

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.