求A^x + B^y = C^z的min（A，B，C）/ gcd（A，B，C）的最大值
The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z
摘要：
本文回答了小埃德·佩格提出的一个问题。2001年：“对于A^x + B^y = C^z，其中A，B，C >= 1; x，y，z >= 3，min（A，B，C）/ gcd（A，B，C）的最大值是多少？分析了这种形式的方程，展示了它们如何映射到具有互质基的指数丢番图方程。提供搜索算法以在给定的等式范围内找到最大的min/gcd值。该算法预先计算了一个多千兆字节的幂剩余信息查找表，该查找表用于通过单个阵列查找消除99%以上的输入，而无需任何进一步的计算。在通过该测试的输入上，该算法然后执行进一步的幂剩余测试，通过使用查找到预先计算的表来避免模幂，并且通过使用乘法逆来避免除法。该算法用于显示C^z <= 2^100的所有方程的最大min/gcd值。
英文摘要：
This paper answers a question asked by Ed Pegg Jr. in 2001: "What is the maximal value of min(A,B,C)/ gcd(A,B,C) for A^x + B^y = C^z with A,B,C >= 1; x,y,z >= 3?" Equations of this form are analyzed, showing how they map to exponential Diophantine equations with coprime bases. A search algorithm is provided to find the largest min/gcd value within a given equation range. The algorithm precalculates a multi-gigabyte lookup table of power residue information that is used to eliminate over 99% of inputs with a single array lookup and without any further calculations. On inputs that pass this test, the algorithm then performs further power residue tests, avoiding modular powering by using lookups into precalculated tables, and avoiding division by using multiplicative inverses. This algorithm is used to show the largest min/gcd value for all equations with C^z <= 2^100.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
