On some conjectures of exponential Diophantine equations (2012.15401v1)
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 $$r=2, m\equiv 3(mod 4), m>n{10.4\times10{11}\log(5.2\times10{11}\log n)}.$$