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Arithmetic behaviour of the sums of three squares. (English) Zbl 0623.10035
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 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 R. Schulze-Pillot [Invent. Math. 75, 283–299 (1984; Zbl 0533.10021)].
As a consequence, applying directly a theorem of 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 $$\mathbb Q$$.

##### MSC:
 11P05 Waring’s problem and variants 11R32 Galois theory 11E12 Quadratic forms over global rings and fields 11F11 Holomorphic modular forms of integral weight 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
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##### References:
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