Abstract
In this paper, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=k$, where $k\in\{\pm1,\pm4\}$, which is a Padovan number, with a few exceptions that we completely characterize.
Highlights
Let {Pl}l≥0 be the Padovan sequence given by Pl = Pl−2 + Pl−3, for l ≥ 3, where P0 = 0, P1 and P2 = 1
It is well known that all positive solutions (X, Y ) of (1.1) are given by
It is well know that all positive solutions (X, Y ) of (1.2) are given by
Summary
Let {Pl}l≥0 be the Padovan sequence given by Pl = Pl−2 + Pl−3 , for l ≥ 3 , where P0 = 0 , P1 and P2 = 1. Let d > 1 be a positive integer which is not a perfect square. One expects that the answer to such a question has at most one positive integer solution n for any given d except maybe for a few (finitely many) values of d. We will prove the following theorems: Theorem 1.1 Let d ≥ 2 be square-free. Has at most one solution (m, l) in positive integers with the following exceptions:. We recall a few properties of the Padovan sequence {Pl}l≥0 which are useful in proving our theorem.
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