On the Ramanujan-Nagell type Diophantine equation x2+Akn=B

Abstract

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation x 2 + Ak n = B has at most three nonnegative integer solutions (x, n) for A = 1, 2, 4, k an odd prime and B a positive integer. Therefore, we partially confirm two conjectures of Ulas from [23].