当前位置: X-MOL 学术Period. Math. Hung. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The extension of the $$D(-k)$$ D ( - k ) -pair $$\{k,k+1\}$$ { k , k + 1 } to a quadruple
Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2021-08-21 , DOI: 10.1007/s10998-021-00424-8
Nikola Adžaga 1 , Alan Filipin 1 , Yasutsugu Fujita 2
Affiliation  

Let \(n\ne 0\) be an integer. A set of m distinct positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a D(n)-m-tuple if \(a_ia_j + n\) is a perfect square for all \(1\le i < j \le m\). Let k be a positive integer. In this paper, we prove that if \(\{k,k+1,c,d\}\) is a \(D(-k)\)-quadruple with \(c>1\), then \(d=1\). The proof relies not only on standard methods in this field (Baker’s linear forms in logarithms and the hypergeometric method), but also on some less typical elementary arguments dealing with recurrences, as well as a relatively new method for the determination of integral points on hyperelliptic curves.



中文翻译:

$$D(-k)$$ D ( - k ) -pair $$\{k,k+1\}$$ { k , k + 1 } 扩展为四元组

\(n\ne 0\)是一个整数。一组的不同的正整数\(\ {A_1,A_2,\ ldots,A_M \} \)被称为dÑ) -元组如果\(a_ia_j + N \)是所有完全平方\( 1\le i < j \le m\)。令k为正整数。在本文中,我们证明如果\(\{k,k+1,c,d\}\)\(D(-k)\) -quadruple with \(c>1\),则\( d=1\). 证明不仅依赖于该领域的标准方法(贝克的对数线性形式和超几何方法),还依赖于一些不太典型的处理递归的基本论证,以及一种相对较新的确定超椭圆上积分点的方法曲线。

更新日期:2021-08-21
down
wechat
bug