A Lucas\u2013Lehmer approach to generalised Lebesgue\u2013Ramanujan\u2013Nagell equations

Abstract

We describe a computationally efficient approach to resolving equations of the form C_1x^2 + C_2 = y^n in coprime integers, for fixed values of C_1, C_2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.