An Effective Method for Solving a Class of Transcendental Diophantine Equations

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.