论文标题
在两个未知数中，特殊类型的单位方程的解决方案数ii
Number of solutions to a special type of unit equations in two unknowns, II
论文作者
论文摘要
本文有助于R. Scott和R. Styer的猜想，该猜想断言，对于任何大于1的固定相对主要的正整数$ a，b $ a，b $和$ c $，最多有一个解决方案$ a^x+a^x+b^y = c^y = c^y = c^y = c^z $在正整数中$ x，y $和$ z $除外。基本结果证明了在某些一致性条件模式下的猜想，$ a $和$ b $。由于应用程序确认，如果$ c $采用一些较小的值，包括到目前为止发现的费马特素数，尤其是Scott著名定理的分析证明[R. [R.斯科特，在方程式上$ p^x-b^y = c $和$ a^x+b^y = c^z $，J。数字理论44（1993），第2号，153-165]以纯代数方式解决$ c = 2 $的猜想。该方法可以针对较小的模量案例进行推广，事实证明，对于$ c $的许多特定值而言，猜想是正确的，不是完美的力量。主要的新颖性是将$ p $ - 亚法类似物的一种特殊类型应用于贝克在对数中线性形式的理论，这是由于存在方程式的两个假设解决方案而产生的一定分解性关系。其他工具在复杂情况下包括贝克的理论及其对数字字段的非库赛类似物，以及通过有理数和二次数字以及广泛的计算，以及各种基本论点。
This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo $c$ on $a$ and $b$. As applications the conjecture is confirmed to be true if $c$ takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott [R. Scott, On the equations $p^x-b^y=c$ and $a^x+b^y=c^z$, J. Number Theory 44(1993), no.2, 153-165] solving the conjecture for $c=2$ in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of $c$ not being perfect powers. The main novelty is to apply a special type of the $p$-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation.
