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

Abstract

Let {P n } n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = P 2 = 1, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we find all positive square-free integers d ≥ 2 such that the Pell equations x 2 − dy 2 = , where ∈ {±1, ±4}, have at least two positive integer solutions (x, y) and (x , y ) such that each of x and x is a product of two Padovan numbers.

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

“…and r is minimal such that (U n ) satisfies a relation above, then we say that U = (U n ) = (U n ) n≥0 is a linear recurrence sequence (of integers) of order r. Throughout the paper we always assume that a recurrence sequence is given by its minimal length relation (2). We shall also use the notation…”

“…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.…”