On the number of solutions of the generalized Ramanujan-Nagell equation

Abstract

Let D₁ and D₂ be coprime positive integers and let k be an odd positive integer coprime with D₁D₂. We consider the Diophantine equation D₁x² + D₂ = kⁿ in the unknowns x≥1, n≥1. We give a necessary and sufficient condition on D₁, D₂ and k under which this equation has at most 2^ω(k)−1 solutions where ω(k) denoted the number of distinct prime divisors of k. Thus, under a necessary and sufficient conditon, the equation has at most one solution whenever k is a prime. We also consider some related equations and we prove that the Diophantine equation x²+7=4yⁿ has no solution in integers x≥1, y > 2 and n >1.