In this paper, we consider the Diophantine equation in the title, where p,q are distinct odd prime numbers and a,b are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of p,q,a and b. 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.

中文翻译：

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丢番图方程 x2 + paqb = yq

在本文中，我们考虑标题中的丢番图方程，其中p,q是不同的奇质数和一种,b是自然数。我们给出了许多结果，给出了该方程存在整数解的条件，根据p,q,一种和b. 我们的方法本质上是基本的，并且基于对某些卢卡斯序列的原始除数的研究以及某些多项式的分解。