On some conjectures of exponential Diophantine equations

TL;DR

This paper investigates specific exponential Diophantine equations related to sums of powers of algebraic integers, proving the absence of nontrivial solutions under certain conditions involving parameters r, m, and n.

Contribution

It establishes new nonexistence results for solutions of exponential Diophantine equations with particular algebraic and number-theoretic constraints.

Findings

No nontrivial solutions when r ≡ 2 mod 4 and m > specified bounds.

No solutions for r=2 when m > n^{large logarithmic bound}.

Results extend understanding of exponential Diophantine equations with algebraic integer conditions.

Abstract

In this paper, we consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a, b, c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{r}, r\in Z^{+}, 2\mid r$ with $b$ even. That is $$a=\mid Re(m+n\sqrt{-1})^{r}\mid, b=\mid Im(m+n\sqrt{-1})^{r}\mid, c=m^{2}+n^{2},$$ where $m, n$ are positive integers with $m>n, m-n\equiv1(mod 2),$ gcd$(m, n)=1.$ $(x, y, z)= (2, 2, r)$ is called the trivial solution of the equation. In this paper we prove that the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r\equiv 2(mod 4), m\equiv 3(mod 4), m>\max\{n^{10.4\times10^{11}\log(5.2\times10^{11}\log n)}, 3e^{r}, 70.2nr\}.$$ Especially the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r=2, m\equiv 3(mod 4), m>n^{10.4\times10^{11}\log(5.2\times10^{11}\log n)}.$$