On $X$-coordinates of Pell equations which are repdigits

TL;DR

This paper proves the finiteness of certain Pell equation solutions with $X$-coordinates as base $b$-repdigits and provides an upper bound for the largest such $d$ based on $b$.

Contribution

It establishes the finiteness of non-square $d$ with two solutions having $X$-coordinates as base $b$-repdigits and derives an explicit upper bound for the largest such $d$.

Findings

Finiteness of such $d$ for given $b$.

An explicit upper bound on the largest $d$ in terms of $b$.

Characterization of solutions with $X$-coordinates as repdigits.

Abstract

Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that their $X$-coordinates are base $b$-repdigits. Recall that a base $b$-repdigit is a positive integer all whose digits have the same value when written in base $b$. We also give an upper bound on the largest such $d$ in terms of $b$.