The generalized Ramanujan-Nagell equation x^2+D=2^s p_1^k p_2^l

What is it about?

It has been known that, for a given positive integer D and given primes p_1, p_2, the equation x^2+D=2^s p_1^k p_2^l has only finitely many integer solutions (x, s, k, l) (s=0 or 2). We show that indeed this equation has at most 63 solutions.

Featured Image

Why is it important?

Diophantine equations x^2+D=p^k and x^2+D=4p^k for a given integer D and a given prime p have been extensively studied. A generalized equation x^2+D=y^k for a given integer D also has been studied by several authors. It seems that x^2+D=p_1^{e_1} p_2^{e_2} ... p_r^{e_r} and x^2+D=4p_1^{e_1} p_2^{e_2} ... p_r^{e_r} fr given integer D and given prime p_1, p_2, ..., p_r have been less studied. Evertse has shown that the number of solutions is at most 3.7^{4r+6}, which is fairly large. Our result considerably reduces this bound in the case r=2. Our method will reduce Evertse's bound for larger r.