A note on the diophantine equation x2=4pn-4pm + l2

Abstract

Let l be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation x(2) = 4p(n)-4p(m)+l(2) with l(2) < 4p(m) for any l > 1, where p is a prime, x, m, n are positive integers satisfying gcd(x, l) = 1 and m < n. Meanwhile we give a method to solve the equation with l(2) > 4p(m). As an example of using this method, we find all solutions (p, x, m, n) of the equation for l is an element of {5, 7}.