On the X-coordinates of Pell equations which are products of two Fibonacci numbers

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

“…In this paper, we study a similar problem to that of Kafle, et al [11], but with the Lucas numbers instead of the Fibonacci numbers. That is, we show that there is at most one value of the positive integer x participating in (1.3), which is a product of two Lucas numbers, with a few exceptions that we completely characterize.…”

“…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, . .…”

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

“…In this paper, we study a problem related to that of Kafle et al [14] but with the Padovan sequence instead of the Fibonacci sequence. We also extend the results from the Pell equation (1) in the case = ±1 to the case = ±4.…”

“…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, . .…”

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.…”