In this paper, we show that if u is a fixed binary recurrent sequence of integers whose characteristic equation has real roots and (X_k,Y_k) is the k-th solution of the Pell equation X²-dY²=1 for some non-square integer d>1, the equation Y_k ∈ u has at most two positive integer solutions k provided d exceeds some effectively computable number depending on u.

The authors thank the anonymous referee for pointing out to them reference [4] and Professor Mouhamed Moustapha Fall for useful discussions. Part of this work was done during a very enjoyable visit of F. L. at AIMS Sénégal in February 2018. This author thanks AIMS Sénégal for the hospitality and support. In addition, F. L. was supported in part by Grant CPRR160325161141 of NRF and the Number Theory Focus Area Grant of CoEMaSS at Wits (South Africa) and CGA 17-02804S (Czech Republic).