On $X$-coordinates of Pell equations which are 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 $X^2-d{Y}^2=1$ 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 $b$.

To appear in The Fibonacci Quarterly Journal

Cited by in corpus (1)

• On the $X$-coordinates of Pell equations which are Tribonacci numbers