Arithmetic behaviour of the sums of three squares
type: Item
instance of: scholarly article
title: Arithmetic behaviour of the sums of three squares
zbMATH Open document ID: 0623.10035
DOI: 10.1016/0022-314X(87)90067-9
author: B. George
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: The purpose of this paper is to calculate, for a positive integer \(n\) which has a representation as a sum of three integral squares, the maximum number of summands which can be prime to \(n\). The theorem is that this number is 3 or 2 according to whether \(n\) is coprime to 10, and provided \(n\) is large enough (depending on the radical of \(n\)). The proof uses the evaluation of the number of representations of \(n\) by ternary quadratic forms of special types. This in turn depends on classical results of \textit{C. L. Siegel} [Ann. Math. (2) 36, 527--606 (1935; Zbl 0012.19703)]. Other elements in the proof are Shimura's correspondence and the theory of theta-functions given by \textit{R. Schulze-Pillot} [Invent. Math. 75, 283--299 (1984; Zbl 0533.10021)]. As a consequence, applying directly a theorem of \textit{N. Vila} [Arch. Math. 44, 424--437 (1985; Zbl 0562.12011)], the authors conclude that if \(n\) is congruent to 3 modulo 8 and sufficiently large (as above) then every central extension of the alternating group \(A_n\) can be realized as a Galois group over \(\mathbb Q\).
reviewed by: James Lee Hafner
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 4010619
zbMATH Keywords: sum of three integral squares
zbMATH Keywords: maximum number of summands
zbMATH Keywords: number of representations
zbMATH Keywords: ternary quadratic forms
zbMATH Keywords: Shimura's correspondence
zbMATH Keywords: theta-functions
zbMATH Keywords: central extension
zbMATH Keywords: alternating group
zbMATH Keywords: Galois group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4010619