International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 11, Pages 695-699
doi:10.1155/S0161171202106107

On the Diophantine equation x2+p2k+1=4yn

S. Akhtar Arif1 and Amal S. Al-Ali2

1Department of Mathematics, Girls College of Education, P.O. Box 22171, Riyadh 11495, Saudi Arabia
 2Department of Mathematics, Girls College of Education, P.O. Box 56778, Riyadh 11564, Saudi Arabia

Abstract

It has been proved that if p is an odd prime, y>1, k≥0, n is an integer greater than or equal to 4, (n,3h)=1 where h is the class number of the field Q(−p), then the equation x2+p2k+1=4yn has exactly five families of solution in the positive integers x, y. It is further proved that when n=3 and p=3a2±4, then it has a unique solution k=0, y=a2±1.