A Note on the Number of Solutions of the Generalized Ramanujan-Nagell Equation D_1x~2+D_2=4p~n

摘要

Let D1,D2 be positive integers with 2D1D2 and gcd(D1, D2)=1. Let p be a prime with pD1D2. In this note we prove that the generalized Ramanujan–Nagell equation D1x2+D2=4pn has 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).