Arithmetic progressions in the solution sets of norm form equations
type: Item
instance of: scholarly article
title: Arithmetic progressions in the solution sets of norm form equations
zbMATH Open document ID: 1204.11064
DOI: 10.1216/RMJ-2010-40-2-383
author: Attila Bérczes
author: Lajos Hajdu
author: Attila Pethő
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-06-14T00:00:00Z
review text: Let \(K\) be an algebraic number field of degree \(k\), and let \(\alpha_1,\ldots,\alpha_n\) be linearly independent elements of \(K\) over \(\mathbb Q\). Let \(m\) be a nonzero integer and consider the norm form equation \[ N_{K/\mathbb Q}(x_1\alpha_1+\dots+x_n\alpha_n)=m \] in integers \(x_1,\ldots,x_n\). Let \(H\) denote the solution set of the equation above. Several arithmetic properties of the elements of \(H\) were studied by Everest, Győry, Mignotte and Shorey. In this paper the authors are concerned with arithmetic progressions in \(H\). Arrange the elements of \(H\) in an \(|H|\times n\) array \(\mathcal H\). Two natural questions appeared about arithmetical progressions. The ``horizontal'' one: do there infinitely many rows of \(H\) exist which form arithmetic progressions; and ``vertical'' one: do arbitrary long arithmetic progressions in some column of \(\mathcal H\) exist? The ``horizontal'' problem was treated by Bérczes, Pethő and later Bérczes, Pethő, Ziegler, Bazsó computed horizontal solutions. The main goal of this paper is to generalize the result of Pethő and Ziegler to arbitrary norm form equations. The authors prove that the length of any arithmetic progressions in \(H\) is bounded. Their next theorem shows that in general if \(H\) contains algebraic arithmetic progressions at all, then it contains infinitely many. Moreover, they generalize the following result (Newman): the length of arithmetic progressions consisting of units of an algebraic number field of degree \(k\) is at most \(k\).
reviewed by: Péter Olajos
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 5720611
zbMATH Keywords: norm form equations
zbMATH Keywords: arithmetic progressions
MaRDI profile type: Publication
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full work available at URL: rmj-2010-40-2-383
OpenAlex ID: W2002511514
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description: scientific article; zbMATH DE number 5720611