虚二次数环中丢番图m元组的大小

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2018-07-05 DOI:10.1142/S1664360719500206

Nikola Advzaga

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引用次数: 10

摘要

丢番图[公式:见文本]元组是一组不同的整数，使得任意两个不同的元素加一的乘积是完全平方。最近证明了在正整数中不存在丢番图五元组。我们在虚二次域的整数环上研究了同样的问题。通过使用由丢番图近似证明的间隙原理，我们表明[公式:见文本]。与正整数中类似结果的证明相比，我们的证明相对简单。

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On the size of Diophantine m-tuples in imaginary quadratic number rings

A Diophantine [Formula: see text]-tuple is a set of [Formula: see text] distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that [Formula: see text]. Our proof is relatively simple compared to the proofs of similar results in positive integers.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.