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