Pell and Pell–Lucas numbers of the form $$x^a\pm x^b+1$$

Abstract

Let P k and Q k be the k th Pell and Pell-Lucas terms of the Pell sequence fP n g n ! 0 and the Pell-Lucas sequence fQ n g n ! 0 , respectively. In this paper, we study the Diophantine equations P n ¼ x a AE x b þ 1 and Q n ¼ x a AE x b þ 1, in positive integers (n, x, a, b) and determine the explicit upper bounds for n. We also completely solve these equations in positive integers (n, x, a, b) with 0 b\a and 2 x 20.

A New Family of Number Sequences: Leonardo-Alwyn Numbers

“…Leonardo Fibonacci and Alwyn Horadam examined number sequences defined by recurrence relations, which were then studied over the years (see, example, [4,10,19,22,25,27,28]). The Leonardo sequence, also known as Leonardo numbers, is a linear recurrent sequence of integers related to the Fibonacci sequence (see [30]).…”

On the Diophantine Equation Fn = x^a \pm x^b \pm 1 in Mersenne and Fermat Numbers

“…Laishram and Luca [7] studied a more general Diophantine equation F n = x a ± x b ± 1 with x composed of two prime divisors, showing that it has only finitely many positive integer solutions (n, x, a, b) with max{a, b} ≥ 2. Recently, Kafle, Rihane and Togbé [6] have investigated about Pell and Pell-Lucas numbers (instead of Fibonacci numbers) of the form x a ± x b + 1, completely solving this equation for each x ∈ [2,20].…”