On px2+q2n=yp and related Diophantine equations

The title equation, where p>3 is a prime number ≢7(mod8), q is an odd prime number and x, y, n are positive integers with x, y relatively prime, is studied. When p≡3(mod8), we prove (Theorem 2.3) that there are no solutions. For p≢3(mod8) the treatment of the equation turns out to be a difficult task. We focus our attention to p=5, by reason of an article by F. Abu Muriefah, published in J. Number Theory 128 (2008) 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equation 5x2−4=yn (integer solutions), interesting in themselves, which are exposed in Sections 3 and 4. These last results are obtained by using tools such as linear forms in two logarithms and hypergeometric series.