On the $x$--coordinates of Pell equations that are products of two Pell numbers

Let $ (P_m)_{m\ge 0} $ be the sequence of Pell numbers given by $ P_0=0, ~ P_1=1 $, and $ P_{m+2}=2P_{m+1}+P_m $ for all $ m\ge 0 $. In this paper, for an integer $d\ge 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2} =\pm 1$, which is a product of two Pell numbers.