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 $m>n,m-n\equiv 1(mod2),$ 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 $$r\equiv 2(mod4),m\equiv 3(mod4),m>\max \{n^{10.4\times {10}^{11}\log (5.2\times {10}^{11}\log n)},3e^r,70.2nr\}\mathrm{.}$$ Especially the equation has no nontrivial solutions in positive integers $x, y, z$when$$r=2, m\equiv 3(mod 4), m>n^{10.4\times10^{11}\log(5.2\times10^{11}\log n)}.$$