论文标题
两个变量中特殊类型的单位方程的解决方案数量
Number of solutions to a special type of unit equations in two variables
论文作者
论文摘要
对于任何带有$ \ min \ {a，b，b，c \}> 1 $的固定副本正整数$ a，b $和$ c $，我们证明了方程$ a^x+b^y = c^z $最多在正整数中具有两个解决方案$ x，y $ $ $ $ z $，除了一个特定的情况下，这是一个特定的解决方案。我们的结果本质上是敏锐的，因为有很多例子允许方程在正整数中具有两个解决方案。从Fermat方程的众所周知的概括的角度来看，它也被视为对贝内特（M.A. Bennett）在CASAD S.S.S.S.Pillai的某些指数方程式中对著名定理的3变量概括。 J. Math。 53（2001），第2、897--922号]断言Pillai的类型方程$ a^x-b^y = c $最多具有正整数中的最多两个解决方案$ x $和$ y $，用于任何固定的正整数$ a，b $ a，b $和$ c $，带有$ \ min \ min \ min \ {a，b \ \}> 1 $。
For any fixed coprime positive integers $a,b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x,y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equations of S.S.Pillai, Canad. J. Math. 53(2001), no.2, 897--922] which asserts that Pillai's type equation $a^x-b^y=c$ has at most two solutions in positive integers $x$ and $y$ for any fixed positive integers $a,b$ and $c$ with $\min\{a,b\}>1$.
