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
Girls College of Education, Science Sections (Mathematics), Sitten Street, Al Malaz, P.O. Box 27104, Riyadh, Saudi Arabia
Received 6 May 1999; Revised 20 February 2000
Copyright © 2001 Fadwa S. Abu Muriefah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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)
.