The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z

TL;DR

This paper investigates the maximum ratio of min(A,B,C) to gcd(A,B,C) for solutions of A^x + B^y = C^z with exponents ≥ 3, introducing an efficient search algorithm with large lookup tables to analyze equations up to C^z ≤ 2^100.

Contribution

It presents a novel, highly efficient algorithm utilizing large precomputed tables to identify maximal min/gcd ratios in exponential Diophantine equations of a specific form.

Findings

Identified the maximal min/gcd ratio for equations with C^z ≤ 2^100.

Developed a lookup table-based algorithm reducing computation time by over 99%.

Mapped the problem to exponential Diophantine equations with coprime bases.

Abstract

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.