Arithmetic behaviour of the sums of three squares
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 \(\mathbb Q\).
scientific article; zbMATH DE number 4010619