On the number of extensions of a Diophantine triple
On the number of extensions of a Diophantine triple
复制标题
关于丢番图三元组的扩张数
DOI：
10.1142/s1793042118500549
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发表时间：
2018
影响因子：
0.7
通讯作者：
Takafumi Miyazaki
中科院分区：
文献类型：
--
作者：
Mihai Cipu;Yasutsugu Fujita;Takafumi Miyazaki
如果一组正整数中任意两个元素的积为完全平方，则称为丢番图元组。任何丢番图三元组被推测为唯一地扩展为丢番图四元组，通过连接一个超过三元组中最大元素的元素。第二和第三位作者先前的工作表明，固定丢番图三重的这种扩展的数量最多为11。在本文中，我们证明了这个数最多为8。
A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.