Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations

Abstract

It is well-known that for p = 1, 2, 3,7,11,19, 43, 67, 163 the class number of Q( √ −p) is one. We use this fact to determine all the solutions of x 2 + p m = 4y n in non-negative integers x, y, m and n except for p = 2. For this value of p, we obtain all the solutions (x, y, m, n) assuming the conjecture of S. A. Arif and F. S. Abu Muriefah [3].