On the Diophantine equation N X^2 + 2^L 3^M = Y^N

TL;DR

This paper proves the non-existence of solutions to a specific exponential Diophantine equation involving powers of 2 and 3, extending previous results and employing advanced number theory techniques.

Contribution

It generalizes prior work by proving the absence of solutions for a broader class of Diophantine equations with specific coprimality conditions.

Findings

No solutions exist for the given equation under specified conditions.

The proof utilizes results on defective Lehmer pairs.

Extends previous non-existence results to a more general setting.

Abstract

We prove that the Diophantine equation N X^2 + 2^L 3^M = Y^N has no solutions (N,X,Y,L,M) in positive integers with N > 1 and gcd(NX,Y) = 1, generalizing results of Luca, Wang and Wang, and Luca and Soydan. Our proofs use results of Bilu, Hanrot, and Voutier on defective Lehmer pairs.