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

Abstract

Let $$P_k$$ and $$Q_k$$ be the $$k^{\text{ th }}$$ Pell and Pell–Lucas terms of the Pell sequence $$\{P_n\}_{n\ge 0}$$ and the Pell–Lucas sequence $$\{Q_n\}_{n\ge 0}$$ , respectively. In this paper, we study the Diophantine equations $$P_n = x^a\pm x^b+1$$ and $$Q_n = x^a\pm x^b+1$$ , 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 $$0\le b < a$$ and $$2\le x \le 20$$ .