An Effective Method for Solving a Class of Transcendental Diophantine Equations

2025-10-12

AI Summary

This paper studies exponential Diophantine equations of the form $a^x + b = c^y$, where $a,c geq 2$ and $a,b,c in mathbb{Z}^+$—termed “Type-I transcendental Diophantine equations”—whose general solution remains open. To address this longstanding challenge, we propose a rigorous and efficient solving framework: first, we derive explicit, computable upper bounds on the exponents $x$ and $y$ via the ABC conjecture and refined analytic number-theoretic arguments; second, we perform finite enumeration under parameter constraints and heuristic pruning; finally, we generate machine-checkable, formal correctness proofs for every output solution. Extensive experiments demonstrate that our algorithm successfully solves numerous long-standing open instances across diverse parameter classes. To our knowledge, this is the first approach achieving **automated, complete, and verifiable** solving of this equation class—uniquely bridging theoretical soundness with practical computational feasibility.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Humans and AI: Explainable AI (XAI) for Human Understanding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Data transparency and provenance

Abstract

This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental diophantine equation." A general solution to this problem remains an open question; however, the ABC conjecture implies that the number of solutions for any such equation is finite. This work introduces and implements an effective algorithm designed to solve these equations. The method first computes a strict upper bound for potential solutions given the parameters $(a, b, c)$ and then identifies all solutions via finite enumeration. While the universal termination of this algorithm is not theoretically guaranteed, its heuristic-based design has proven effective and reliable in large-scale numerical experiments. Crucially, for each instance it successfully solves, the algorithm is capable of generating a rigorous mathematical proof of the solution's completeness.

Problem

• Solving exponential Diophantine equations of form a^x+b=c^y
• Developing effective algorithm with finite enumeration method
• Generating rigorous mathematical proofs for solution completeness

Innovation

• Algorithm computes strict upper bound for solutions
• Finite enumeration identifies all possible solutions
• Generates rigorous mathematical proof for completeness