$Y$-coordinates of Pell equations in binary recurrences

Let $d>1$ be an integer which is not a square and $(X_n,Y_n)$ be the $n$th solution of the Pell equation $X^2-dY^2=\pm 1$. Given an interesting set of positive integers $U$, we ask how many positive integer solutions $n$ can the equation $Y_n\in U$ have. We show that under mild assumptions on $U$ (for example, when $1\in U$ and $U$ contains infinitely many even integers), then the equation $Y_n\in U$ has two solutions $n$ for infinitely many $d$. We show that this is best possible whenever $U$ is the set of values of a binary recurrent sequence $\{u_m\}_{m\ge 1}$ with real roots and $d$ is large enough (with respect to $U$). We also show that for the particular case when $u_m=2^m-1$, the equation $Y_n=2^m-1$ has at most two positive integer solutions $(n,m)$ for all $d$. The proofs use linear forms in logarithms. This is joint work with Bernadette Faye.