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

Abstract

Let {Ln} n≥0 be the sequence of Lucas numbers given by L0 = 2, L1 = 1 and Ln+2 = Ln+1 + Ln for all n ≥ 0. In this paper, for an integer d ≥ 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 = ±1 which is a product of two Lucas numbers, with a few exceptions that we completely characterize.

2-Peble Triangles Over Figurate Numbers

“…The articles that use Lucas numbers (1) , x-coordinate of Pell's equation with powers of 2 (2) , and Fibonacci numbers (3) in Pell equations served as inspirations for this article. New types of rectangles and triangles are defined and some of their properties are explored by using Diophantine equations in articles (4,5) .…”

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.…”

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

“…This is sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [21]. The first few terms of this sequence are 3,4,5,7,9,12,16,21,28,37,49,65,86,114,151, . .…”

“…In this paper, we let U := {P n P m : n ≥ m ≥ 0} be the sequence of products of two Padovan numbers. The first few members of U are U = {0, 1, 2, 3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25, 27, 28, 32, 35, . .…”

“…Several other related problems have been studied where x l belongs to some interesting positive integer sequences. For example, see [2,3,6,7,8,9,10,12,15,16,17,18,19].…”