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Regular ternary triangular forms. (English) Zbl 1446.11058

A polynomial of the type \[ \Delta:=\Delta(a_1,a_2,a_3)=a_1\frac{x_1(x_1+1)}{2}+a_2\frac{x_2(x_2+1)}{2}+a_3\frac{x_3(x_3+1)}{2}, \] where \(a_1,a_2,a_3\) are fixed positive integers, is called a ternary triangular form. Such a form is primitive if g.c.d.\((a_1,a_2,a_3)=1\). An integer \(N\) is represented by \(\Delta\) if there exist \(x_1,x_2,x_3\in \mathbb Z\) such that \(\Delta(x_1,x_2,x_3)=N\), and \(N\) is locally represented by \(\Delta\) if there exist such \(x_1,x_2,x_3 \in \mathbb Z_p\) for every prime \(p\). The form \(\Delta\) is regular if it represents all those integers that it locally represents. W. K. Chan and B.-K. Oh [Contemp. Math. 587, 31–46 (2013;Zbl 1298.11026)] proved that there exist only finitely many regular primitive ternary triangular forms. In the present paper, the authors prove that there are exactly 49 triples \((a_1,a_2,a_3)\) of positive integers with \(a_1\leq a_2\leq a_3\) and g.c.d.\((a_1,a_2,a_3)=1\) for which \(\Delta(a_1,a_2,a_3)\) is regular. The methods used in the proof are purely arithmetic.

MSC:

11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11D85 Representation problems

Citations:

Zbl 1298.11026
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References:

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