Kim, Mingyu; Oh, Byeong-Kweon Regular ternary triangular forms. (English) Zbl 1446.11058 J. Number Theory 214, 137-169 (2020). 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. Reviewer: Andrew G. Earnest (Carbondale) Cited in 1 ReviewCited in 3 Documents 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 Keywords:representations of ternary quadratic forms; triangular numbers; ternary triangular forms; regular forms Citations:Zbl 1298.11026 PDFBibTeX XMLCite \textit{M. Kim} and \textit{B.-K. Oh}, J. Number Theory 214, 137--169 (2020; Zbl 1446.11058) Full Text: DOI arXiv References: [1] Benham, J. W.; Hsia, J. S., Spinor equivalence of quadratic forms, J. Number Theory, 17, 337-342 (1983) · Zbl 0532.10012 [2] Burgess, D. A., On character sums and L-series, II, Proc. Lond. Math. Soc., 13, 524-536 (1963) · Zbl 0123.04404 [3] Chan, W. K.; Oh, B.-K., Representations of Integral Quadratic Polynomials, Contemp. Math., vol. 587, 31-46 (2013), Am. Math. Soc. · Zbl 1298.11026 [4] Chan, W. K.; Ricci, J., The representation of integers by positive ternary quadratic polynomials, J. Number Theory, 156, 75-94 (2015) · Zbl 1395.11043 [5] Dickson, L. E., Ternary quadratic forms and congruences, Ann. Math., 28, 331-341 (1927) · JFM 53.0133.03 [6] Dickson, L. E., History of the Theory of Numbers, vol. II (1999), AMS Chelsea Publ. [7] Earnest, A. G., The representation of binary quadratic forms by positive definite quaternary quadratic forms, Trans. Am. Math. Soc., 345, 853-863 (1994) · Zbl 0810.11019 [8] Earnest, A. G., An application of character sum inequalities to quadratic forms, (Number Theory. Number Theory, Canadian Math. Soc. Conference Proceedings, vol. 15 (1995)), 155-158 · Zbl 0833.11012 [9] Jagy, W. C.; Kaplansky, I.; Schiemann, A., There are 913 regular ternary forms, Mathematika, 44, 332-341 (1997) · Zbl 0923.11060 [10] Jones, B. W.; Pall, G., Regular and semi-regular positive ternary quadratic forms, Acta Math., 70, 165-190 (1940) · JFM 65.0141.02 [11] Kim, B. M.; Kim, M.-H.; Oh, B.-K., 2-universal positive definite integral quinary quadratic forms, Contemp. Math., 249, 51-62 (1999) · Zbl 0955.11011 [12] Kim, M.; Oh, B.-K., The number of representations by a ternary sum of triangular numbers, J. Korean Math. Soc., 56, 67-80 (2019) · Zbl 1456.11038 [13] Kitaoka, Y., Arithmetic of Quadratic Forms (1993), Cambridge University Press · Zbl 0785.11021 [14] Lemke Oliver, R. J., Representation by ternary quadratic forms, Bull. Lond. Math. Soc., 46, 1237-1247 (2014) · Zbl 1304.11019 [15] O’Meara, O. T., Introduction to Quadratic Forms (1963), Springer Verlag: Springer Verlag New York · Zbl 0107.03301 [16] Oh, B.-K., Regular positive ternary quadratic forms, Acta Arith., 147, 233-243 (2011) · Zbl 1241.11044 [17] Watson, G. L., Some problems in the theory of numbers (1953), University of London, Ph.D. Thesis This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.