International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 6, Pages 373-381
doi:10.1155/S0161171201004835

On the Diophantine equation Ax2+22m=yn

Abstract

Let h denote the class number of the quadratic field ℚ(−A) for a square free odd integer A>1, and suppose that n>2 is an odd integer with (n,h)=1 and m>1. In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with x and y odd. The case n=3 is solved completely. A result of E. Brown for A=3 is improved and generalized to the case where A is a prime ≢7(mod8) .