论文标题
关于关于$ a^x+b^y = c^z $，ii的解决方案数量的猜想
On a conjecture concerning the number of solutions to $a^x+b^y=c^z$, II
论文作者
论文摘要
令$ a $，$ b $，$ c $是$ a <b $的不同素数。令$ s（a，b，c）$表示等式$ a^x + b^y = c^z $的正整数解决方案$（x，y，z）的数量。在上一篇论文\ cite {lest}中，表明如果$（a，b，c）$是$ s（a，b，c）> 1 $和$（a，b，b，c）$的三重不同的素数，那么$（a，b，c）$不是$（a，b，c）$的六个已知三个这样的三个这样的三倍之一。在本文中，我们消除了其中两种情况（使用其中一种情况的某些持续分数的特殊特性，并使用Dirichlet的四分之一残基的结果）。然后，我们表明剩余案例需要严格的限制，包括以下限制：$ a = 2 $，$ b \ equiv 1 \ bmod 48 $，$ c \ equiv 17 \ mod 48 $，$ b> 10^9 $，$ c> 10^{18} $; $ u_c（b）$或$ u_b（c）$中的至少一个必须是奇数（其中$ u_p（n）$是最小整数$ t $，因此$ n^t \ equiv 1 \ equiv 1 \ bmod p $）;除一种特定情况外，2必须是八粒残留模式$ c $； $ 2 \ MID V_2（B-1）\ LE V_2（C-1）$（其中$ v_2（n）$满足$ 2^{v_2（n）} \ Parallel n $）;必须有两个解决方案$（x_1，y_1，z_1）$和$（x_2，y_2，z_2）$，$ 1 = z_1 <z_1 <z_2 $，$ x_1 \ ge 28 $或$ x_2 \ ge 88 $。这些结果支持在\ cite {scst6}中提出的猜想，并在\ cite {lest}中改善结果。
Let $a$, $b$, $c$ be distinct primes with $a<b$. Let $S(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. In a previous paper \cite{LeSt} it was shown that if $(a,b,c)$ is a triple of distinct primes for which $S(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then $(a,b,c)$ must be one of three cases. In the present paper, we eliminate two of these cases (using the special properties of certain continued fractions for one of these cases, and using a result of Dirichlet on quartic residues for the other). Then we show that the single remaining case requires severe restrictions, including the following: $a=2$, $b \equiv 1 \bmod 48$, $c \equiv 17 \mod 48$, $b > 10^9$, $c > 10^{18}$; at least one of the multiplicative orders $u_c(b)$ or $u_b(c)$ must be odd (where $u_p(n)$ is the least integer $t$ such that $n^t \equiv 1 \bmod p$); 2 must be an octic residue modulo $c$ except for one specific case; $2 \mid v_2(b-1) \le v_2(c-1)$ (where $v_2(n)$ satisfies $2^{v_2(n)} \parallel n$); there must be exactly two solutions $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ with $1 = z_1 < z_2$ and either $x_1 \ge 28$ or $x_2 \ge 88$. These results support a conjecture put forward in \cite{ScSt6} and improve results in \cite{LeSt}.
