通过邻接较小元素扩展 Diophantine 三元组 II

IF 0.6 3区 数学 Q3 MATHEMATICS
Mihai Cipu, Andrej Dujella, Yasutsugu Fujita
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引用次数: 0

摘要

让 \(\{a_1,b,c\}\) 和 \(\{a_2,b,c\}\) 是二叉三元组,有 \(a_1<b<a_2<c\) 和 \(a_2\ne b+c-2sqrt{bc+1}\).把(d_2=a_2+b+c+2a_2bc-2r_2st),其中(r_2=sqrt{a_2b+1}),(s=sqrt{ac+1})和(t=sqrt{bc+1})。在本文中,我们证明如果(c \le 16\mu ^2 b^3),其中(\mu =\min \{a_1,d_2\}),那么(\{a_1,a_2,b,c\})就是一个二重四元数。把这个结果和我们之前的一个结果结合起来,就意味着如果 \(\{a_i,b,c,d\}\) \((i/in/{1,2,3/})\)是具有 \(a_1<a_2<b<a_3<c<d\) 的二重四次方,那么 \(a_3=b+c-2(sqrt{bc+1})。随即可以得出,不存在一个七元组 \(\{a_1,a_2,a_3,a_4,b,c,d\}) with \(a_1<a_2<;b<a_3<a_4<c<d\) such that \(\{a_i,b,c,d\}\) \((i \in \{1,2,3,4\})\) are Diophantine quadruples.此外,还证明了只有有限多个六次元 \(\{a_1,a_2,a_3,b,c,d\}) with \(a_1<b<a_2<a_3<c<d\) such that \(\{a_i,b,c,d\}) \((i \in \{1,2,3\})\) are Diophantine quadruples.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Extensions of a Diophantine triple by adjoining smaller elements II

Let \(\{a_1,b,c\}\) and \(\{a_2,b,c\}\) be Diophantine triples with \(a_1<b<a_2<c\) and \(a_2\ne b+c-2\sqrt{bc+1}\). Put \(d_2=a_2+b+c+2a_2bc-2r_2st\), where \(r_2=\sqrt{a_2b+1}\), \(s=\sqrt{ac+1}\) and \(t=\sqrt{bc+1}\). In this paper, we prove that if \(c \le 16\mu ^2 b^3\), where \(\mu =\min \{a_1,d_2\}\), then \(\{a_1,a_2,b,c\}\) is a Diophantine quadruple. Combining this result with one of our previous results implies that if \(\{a_i,b,c,d\}\) \((i\in \{1,2,3\})\) are Diophantine quadruples with \(a_1<a_2<b<a_3<c<d\), then \(a_3=b+c-2\sqrt{bc+1}\). It immediately follows that there does not exist a septuple \(\{a_1,a_2,a_3,a_4,b,c,d\}\) with \(a_1<a_2<b<a_3<a_4<c<d\) such that \(\{a_i,b,c,d\}\) \((i \in \{1,2,3,4\})\) are Diophantine quadruples. Moreover, it is shown that there are only finitely many sextuples \(\{a_1,a_2,a_3,b,c,d\}\) with \(a_1<b<a_2<a_3<c<d\) such that \(\{a_i,b,c,d\}\) \((i \in \{1,2,3\})\) are Diophantine quadruples.

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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
67
审稿时长
>12 weeks
期刊介绍: Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica. Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.
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