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

Abstract

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.

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

“…There are many other researchers who have studied related problems involving the intersection sequence {x n } n≥1 with linear recurrence sequences of interest. For example, see [4,8,7,9,12,13,14,16,17,20].…”

“…This is sequence A000032 on the On-Line Encyclopedia of Integer Sequences (OEIS) [19]. The first few terms of this sequence are 3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571, . .…”

Terms of recurrence sequences in the solution sets of generalized Pell equations

“…We mention a few such recent results; the interested reader may consult their references. In the papers [1,2,3,4,5,6,8,11,15,16,19] the authors provide various finiteness results concerning the values (or sums or products of values) of certain concrete recurrence sequences (such as Fibonacci, Tribonacci, generalized Fibonacci, Lucas, Padovan, Pell, repdigits) in the x coordinate of equation (1), for the cases t = ±1, ±4. Concerning the y-coordinate, we are aware only of two related results.…”