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The first nontrivial genus of positive definite ternary forms. (English) Zbl 0826.11015

The author considers the genus of ternary quadratic forms consisting of the classes of the forms \(f= x^2+ y^2+ 7z^2\) and \(g= x^2+ 2y^2+ 2yz+ 4z^2\). Several theorems are proved regarding which of the integers represented by this genus are represented by \(f\), which by \(g\), and which by both \(f\) and \(g\). For the positive integers not exceeding 100,000 which are represented by the genus but are not covered by any of these theorems, the author reports the result of computations determining which of these integers fail to be represented by each of the forms.

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
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References:

[1] Burton W. Jones, The regularity of a genus of positive ternary quadratic forms, Trans. Amer. Math. Soc. 33 (1931), no. 1, 111 – 124. · JFM 57.0195.03
[2] Burton W. Jones and Gordon Pall, Regular and semi-regular positive ternary quadratic forms, Acta Math. 70 (1939), no. 1, 165 – 191. · Zbl 0020.10701 · doi:10.1007/BF02547347
[3] Gordon Pall, An almost universal form, Bull. Amer. Math. Soc. 46 (1940), 291. · Zbl 0024.24903
[4] -, Representations by quadratic forms, Canad. J. Math. 1 (1949), 344-364. · Zbl 0034.02201
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