Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

TL;DR

This paper completely solves a specific Lebesgue-Ramanujan-Nagell type equation by determining all integer solutions using advanced number theory techniques involving primitive divisors and elliptic curves.

Contribution

It provides the first complete solution to this class of equations, applying modern methods to find all solutions.

Findings

All solutions to the equation are explicitly determined.

The solution relies on primitive divisor results and elliptic curve analysis.

The methods can be applied to similar exponential Diophantine equations.

Abstract

We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\geq 0, n \geq 3$ and $x, y\geq 1$ are unknown integers with $\gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.