Regular ternary triangular forms (Q2187123)

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. \textit{W. K. Chan} and \textit{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.