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.

翻译：本文研究形如 $a^x+b=c^y$ 的指数丢番图方程，其中 $a, b, c$ 为给定的正整数且满足 $a,c \ge 2$，$x,y$ 为正整数未知量。我们将此形式定义为“I型超越丢番图方程”。该问题的一般解仍是一个开放性问题；然而，ABC猜想意味着此类方程的解的数量是有限的。本研究提出并实现了一种用于求解此类方程的有效算法。该方法首先根据参数 $(a, b, c)$ 计算潜在解的严格上界，然后通过有限枚举确定所有解。尽管该算法的普遍终止性在理论上无法保证，但其基于启发式的设计在大规模数值实验中已被证明是有效且可靠的。至关重要的是，对于算法成功求解的每个实例，它都能够生成关于解完备性的严格数学证明。