丢番图m元数组猜想的研究

丢番图m元数组的可扩张性问题是一个历史悠久的数论问题. Diophantus, Fermat, Euler, Baker 等人对其进行过研究. 目前已知任意的丢番图3元组均可扩张为4元， 而人们猜测丢番图5元数组不存在.2004年, Dujella 证明了最大元素充分大的丢番图4元数组均不能扩张成5元.在该问题及相关问题的研究中，申请人已有部分成果，如解决了由Dujella提出的D(-1)猜想中的一类情形.本项目拟通过Baker方法和丢番图逼近方法，研究丢番图m元数组的扩张性.研究内容的主要创新点为：（1）改进该问题目前的最佳上界.（2）力争解决丢番图m元数组的猜想.（3）研究高次和代数整环中的对应问题.

The extensibility of Diophantine m tuples is an old classical problem in Number Theory. Diophantus, Fermat, Euler, Baker, etc. are interested in this problem. It is known that every Diophantine triple can be extend to a Diophantine quadruple. But non Diophantine quintuple be found. A folklore conjecture is that there does not exist a Diophantine quintuple. In 2004, Dujela proved that if the bigest integer in a Diophantine quadruple is sufficiently large, then the conjecture is true. We were obtianed some results in this conjecture and related questions. For example, we solved a problem in D(-1) conjecture. In this project, we will use Baker's method and Diophantine approximation, study the extensibility of Diophantine m tuples. Innovative points of this research mainly include: (1) We will sharp bound of Diophantine quintuples.（2）We will hard work to complete solve Diophantine m tuples Conjecture. (3)Moreover ,we will researh some generalized problem with high degree and corresponding problem in algebraic integer rings.