The diophantine equation x2 + paqb = yq

Abstract

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

On the Diophantine Equation $$dx^2+p^{2a}q^{2b}=4y^p$$

“…d = 2 a 11 b 19 c ). In [12,23], the equation (1.1) has been solved when d = 2 a p b with odd prime p. Recently, Godinho and Neumann [15] obtained some conditions for the existence of the solutions of the following variation:…”