论文标题
关于$ a^x+b^y = c^z $的解决方案数量的猜想
On a conjecture concerning the number of solutions to $a^x+b^y=c^z$
论文作者
论文摘要
令$ a $，$ b $，$ c $确定为$ \ min \ {a，b，c \}> 1 $。令$ n（a，b，c）$表示等式$ a^x + b^y = c^z $的正整数解决方案$（x，y，z）$。我们表明，如果$（a，b，c）$是$ n（a，b，c）> 1 $ and $（a，b，c）$的三重不同的素数，那不是$ a <b $，我们必须拥有$ a = 2 $，$（b，c） \ bmod 24 $和$（a，b，c）$必须满足进一步的强限制，包括$ c> 10^{14} $。这些结果支持了最后两位作者的猜想。
Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{ a,b,c \} >1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then, taking $a<b$, we must have $a=2$, $(b,c) \equiv (1,17)$, $(13,5)$, $(13, 17)$, or $(23, 17) \bmod 24$, and $(a,b,c)$ must satisfy further strong restrictions, including $c>10^{14}$. These results support a conjecture of the last two authors.
