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

Mahadi Ddamulira

Published 2019-06-13Version 1

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 $d\geq 2$ 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.