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 $d\ge 2$ which is square-free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-d{y}^2=\pm 1$ 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 $d\ge 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-d{y}^2=\pm 1$ which is a product of two Pell numbers.

Two papers in one submission for benefit of the reader. The first paper is accepted to appear in The Fibonacci Quarterly. 20+15 pages. arXiv admin note: text overlap with arXiv:1905.11322, arXiv:1803.10434