A Lucas-Lehmer approach to generalised Lebesgue-Ramanujan-Nagell equations

TL;DR

This paper introduces an efficient computational method for solving specific exponential Diophantine equations of the form C_1x^2 + C_2 = y^n, utilizing factorization and the Primitive Divisor Theorem.

Contribution

It presents a novel approach combining factorization techniques and the Primitive Divisor Theorem to solve generalized Lebesgue-Ramanujan-Nagell equations more efficiently.

Findings

Successfully solves a class of equations with fixed coefficients

Demonstrates computational efficiency improvements

Provides a framework applicable to related exponential 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.