On the diophantine equation $x^2 = y^p + 2^k z^p$

Abstract

We attack the equation of the title using a Frey curve, Ribet’s level-lowering theorem and a method due to Darmon and Merel. We are able to determine all the solutions in pairwise coprime integers x,y,z if p≥7 is prime and k≥2. From this we deduce some results about special cases of this equation that have been studied in the literature. In particular, we are able to combine our result with previous results of Arif and Abu Muriefah, and those of Cohn to obtain a complete solution for the equation x 2 +2 k =y n for n≥3.