An application of the BHV theorem to a new conjecture on exponential diophantine equations

TL;DR

This paper applies the BHV theorem on primitive divisors of Lucas numbers to analyze a new conjecture involving exponential Diophantine equations, establishing conditions for the non-existence of solutions.

Contribution

It introduces a novel application of the BHV theorem to a specific exponential Diophantine equation, deriving new non-existence results under certain parameter constraints.

Findings

Proves no positive solutions exist for the equation when A > 8 B^3 and x > z > y.

Establishes that under additional conditions, the only solution is the trivial one (1,1,1).

Extends the understanding of exponential Diophantine equations using advanced number theory.

Abstract

Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A > 8 B^3$, then the equation $(*) \quad (A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$. Combining the above conclusion with some existing results, we can deduce that if $A >8 B^3$ and $B \equiv 2 \bmod 4$, then (*) has only the positive integer solution $(x,y,z) = (1,1,1)$.