关于纯指数三项丢番图方程的研究

Pure ternary exponential Diophantine equation is a family of indeterminate equation whose form is simple and beautiful but whose connotation is deep and rich. It is a challenging basic research topic to determine its positive integer solutions and explore the upper bound of the number of its solutions. Jesmanowicz' conjecture and Terai's conjecture are the core problems that predict the scarcity of the positive integer solutions of these equations. Combining algebraic number theory method, the method of the linear forms in the logarithms, some results in generalized Fermat equations and the method of Diophantine approximation, in this project we will research the number of integer solutions of unit equation in finitely generated groups and improve the upper bound of the number of solutions of general pure ternary exponential Diophantine equation and verify Terai's conjecture and Jesmanowicz' conjecture of some special forms and lay the solid foundation for more deeply researching these conjectures in the future. The research objects of this project include: (1) we will research and determine the positive integer solutions of some pure ternary exponential Diophantine equations of special forms and verify the corresponding conjectures; (2) we will sharpen the upper bound of the number of integer solutions of unit equation in finitely generated rational groups of rank 3 and develop the method of Diophantine approximation furthermore.

纯指数三项丢番图方程是一类形式简单优美、内涵深刻丰富的不定方程，其正整数解的确定及其解数上界的探索是一个富有挑战性的基础研究课题。Jesmanowicz 猜想和 Terai 猜想是其核心问题，预测了这类方程正整数解数目的稀少性。结合代数数论方法、对数线性型方法、广义 Fermat 方程的结果和丢番图逼近方法，本项目拟对有限生成群上单位方程整数解数目的研究，改进一般形式纯指数三项丢番图方程正整数解数目的上界，同时验证某些特殊形式的 Terai 猜想和 Jesmanowicz 猜想，为进一步深入研究这些猜想打下坚实的基础。本项目的研究目标有两个：（1）研究和确定一些特殊形式的纯指数三项丢番图方程的正整数解，验证相应的猜想；（2）削减秩为3的有理数生成群上的单位方程的整数解数目的上界，进一步发展丢番图逼近方法。