On the Diophantine equationAx2+22m=yn

摘要

Lethdenote the class number of the quadratic fieldℚ(−A)for a square free odd integerA>1,and suppose thatn>2is an odd integer with(n,h)=1andm>1. In this paper, it is proved that the equation of the titlehas no solution in positive integersxandyifnhas anyprime factor congruent to 1 modulo 4. Ifnhas no such factor it is proved that there exists at most one solution withxandyodd. The casen=3is solved completely. A result of E.Brown forA=3is improved and generalized to the case whereAis a prime≢7(mod8).