On some conjectures of exponential Diophantine equations

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 $$r=2, m\equiv 3(mod 4), m>n^{10.4\times10^{11}\log(5.2\times10^{11}\log n)}.$$