Diophantine pairs that induce certain Diophantine triples
Diophantine pairs that induce certain Diophantine triples
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诱导某些丢番图三元组的丢番图对
DOI：
10.1016/j.jnt.2019.09.019
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发表时间：
2020
影响因子：
0.7
通讯作者：
Alan Filipin and Yasutsugu Fujita
中科院分区：
文献类型：
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作者：
Mihai Cipu;Alan Filipin and Yasutsugu Fujita
丢番图元组是正整数的集合，其属性是集合中任意两个元素的乘积按单位增加是平方。本文的主要定理表明，任何丢番图三元组，其第二大元素在最小元素的平方和四倍之间，通过加入超过该三元组中最大元素的元素，可以唯一地扩展到丢番图四元组。假设对于某个正整数 T，两个最小元素的形式为 T 2+ 2 T, 4 T 4+ 8 T 3− 4 T，我们会得到类似的结果，这是我们遇到的例外情况。主定理意味着，对于某些正整数 A 和 K∈{1, 2, 3, 4}，具有最小元素 K A 2, 4 K A 4±4 A 的三元组同样有效。
Diophantine tuples are sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. In the main theorem of this paper it is shown that any Diophantine triple, the second largest element of which is between the square and four times the square of the smallest one, is uniquely extended to a Diophantine quadruple by joining an element exceeding the largest element in the triple. A similar result is obtained under the hypothesis that the two smallest elements have the form T 2+ 2 T, 4 T 4+ 8 T 3− 4 T for some positive integer T, which we encounter as an exceptional case. The main theorem implies that the same is valid for triples with smallest elements K A 2, 4 K A 4±4 A for some positive integers A and K∈{1, 2, 3, 4}.