On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation

TL;DR

This paper completely solves a generalized Lebesgue-Ramanujan-Nagell equation by finding all integer solutions using a combination of classical number theory results, computational methods, and analysis of Lehmer sequences.

Contribution

It provides the first complete solution to this generalized equation and extends the approach to related equations using primitive divisor theory.

Findings

All solutions to the generalized equation are explicitly determined.

The method combines primitive divisor results with computational searches.

The approach applies to other Lebesgue-Ramanujan-Nagell type equations.

Abstract

We consider the generalized Lebesgue-Ramanujan-Nagell equation $x^2+17^k41^\ell 59^m=2^\delta y^n$ in the unknown integers $x\geq 1, y>1,n\geq 3$ and $k, \ell, m\geq 0$ satisfying $\gcd(x,y)=1$. We first find all the integer solutions of the above equation, and then use this result to determine all the integer solutions of some other Lebesgue-Ramanujan-Nagell type equations. Our method uses the classical results of Bilu, Hanrot and Voutier on existence of primitive divisors of Lehmer sequences in combination with number theoretic arguments and computer search.