On the $x$--coordinates of Pell equations which are products of two: Lucas numbers, Pell numbers

Abstract: Let $ {L_n}{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. In the first paper, for an integer $d\geq 2$ 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 Lucas numbers, with a few exceptions that we completely characterize. 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 the second paper, for an integer $d\geq 2$ 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.