The x-coordinates of Pell equations and sums of two Fibonacci numbers II

Abstract

Let {F n } n≥0 be the sequence of Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 + F n 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 = ±4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.

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

“…One can easily check that (U n ) is contained in X -this sequence just comes from the even values of m in (6). On the other hand, here a = 6, and (a 2 − 4)/2 = 16 is a full square, yielding that the roots of the characteristic polynomials x 2 − 6x + 1 are units of the ring of integers of Q( √ 2).…”

“…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].…”