For any fixed integers a and b greater than 1, we study the Diophantine equation \(a^x+(ab+1)^y=b^z\). First, we describe a heuristic list of the positive integer solutions x, y and z of the equation. Finally, we solve the equation in some particular cases, which supports the validity of our list of solutions.

中文翻译：

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三个未知数的纯指数丢番图方程

对于任何大于 1 的固定整数a和b，我们研究丢番图方程\ (a ^ x + (ab + 1) ^ y = b ^ z \)。首先，我们描述方程的正整数解x、  y和z的启发式列表。最后，我们在某些特定情况下求解方程，这支持我们的解决方案列表的有效性。