论文标题
关于指数二芬太汀方程的某些猜想
On some conjectures of exponential Diophantine equations
论文作者
论文摘要
在本文中，我们考虑指数二磷酸方程$ a^{x}+b^{y} = c^{z}，其中$ a，b，c $是相对较好的积极整数，使得$ a^{2}+b^{2}+b^{2} = c^{r}，r}，r}，r \ in z^in z^in z^{+b =那是$$ a = \ hid re（m+n \ sqrt {-1}）^{r} \ mid，b = \ mid im（m+n \ sqrt {-1}）^{r} {r} {r} \ mid，c = m^{2}+n^{2}+n^{2}，$ m，$ m，$ m，$ m，$ m，n是$ n是$ n $ n是$ n $ n是$ n $ n $ n是$ n $ ns $ n $ n $ ns mod，m，m，m，m，m。 2），$ gcd $（m，n）=1。$ $ $（x，y，z）=（2，2，r）$称为方程式的微不足道解决方案。在本文中，我们证明该方程在$ x，y，z $中没有非平凡的解决方案，当$ x，y，z $当$$ r \ equiv 2（mod 4），m \ equiv 3（mod 4），m> \ max \ {n^n^{10.4 \ times1010^{11}}}}}}}} 70.2nr \}。$$，尤其是方程式，当$$ r = 2，m \ equiv 3（mod 4），m> n^{10.4 \ times10^{11}} \ log（5.2 \ times10^times10^{11} {11} {11} {11} {11} {11} {11}
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 $$r=2, m\equiv 3(mod 4), m>n^{10.4\times10^{11}\log(5.2\times10^{11}\log n)}.$$
