  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 $b$.
