A Note on the Number of Solutions of the Generalized Ramanujan–Nagell EquationD1x2+D2=4pn Original Research Article

LetD1,D2be positive integers with 2does not divideD1D2and gcd(D1, D2)=1. Letpbe a prime withpdoes not divideD1D2. In this note we prove that the generalized Ramanujan–Nagell equationD1x2+D2=4pnhas at most two positive integer solutions (x, n) except (D1, D2, p)=(1, 7, 2), (3, 5, 2), (1, 11, 3), and (1, 19, 5).