### Electronic Research Archive

2019, 69-87. doi: 10.3934/era.2019010

# Some universal quadratic sums over the integers

• Primary: 11E25; Secondary: 11D85, 11E20

• Let $a,b,c,d,e,f\in\mathbb N$ with $a\geq c\geq e>0$, $b\leq a$ and $b\equiv a\ ({\rm{mod}}\ 2)$, $d\leq c$ and $d\equiv c\ ({\rm{mod}}\ 2)$, $f\leq e$ and $f\equiv e\ ({\rm{mod}}\ 2)$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\Bbb Z$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\Bbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\mathbb Z$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\Bbb Z$ which is related to the form $x^2+y^2+32z^2$.

Citation: Hai-Liang Wu, Zhi-Wei Sun. Some universal quadratic sums over the integers[J]. Electronic Research Archive, 2019, 27: 69-87. doi: 10.3934/era.2019010

### Related Papers:

• Let $a,b,c,d,e,f\in\mathbb N$ with $a\geq c\geq e>0$, $b\leq a$ and $b\equiv a\ ({\rm{mod}}\ 2)$, $d\leq c$ and $d\equiv c\ ({\rm{mod}}\ 2)$, $f\leq e$ and $f\equiv e\ ({\rm{mod}}\ 2)$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\Bbb Z$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\Bbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\mathbb Z$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\Bbb Z$ which is related to the form $x^2+y^2+32z^2$.

 [1] B. C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, RI, 2006. 10.1090/stml/034 2246314 [2] Cassels J. W. S. (1978) Rational Quadratic Forms.Academic Press. [3] Dickson L. E. (1927) Quaternary quadratic forms representing all integers. Amer. J. Math. 49: 39-56. [4] Dickson L. E. (1939) Modern Elementary Theory of Numbers.Univ. of Chicago Press. [5] Earnest A. G. (1980) Congruence conditions on integers represented by ternary quadratic forms. Pacific J. Math. 90: 325-333. [6] Earnest A. G. (1984) Representation of spinor exceptional integers by ternary quadratic forms. Nagoya Math. J. 93: 27-38. [7] Ge F., Sun Z.-W. (2016) On some universal sums of generalized polygonals. Colloq. Math. 145: 149-155. [8] S. Guo, H. Pan and Z.-W. Sun, Mixed sums of squares and triangular numbers (Ⅱ), Integers, 7 (2007), A56, 5pp (electronic). 2373118 [9] Jagy W. C. (1996) Five regular or nearly-regular ternary quadratic forms. Acta Arith. 77: 361-367. [10] Jagy W. C., Kaplansky I., Schiemann A. (1997) There are 913 regular ternary forms. Mathematika 44: 332-341. [11] W. C. Jagy, Integral Positive Ternary Quadratic Forms, Lecture Notes, 2014. Available from: http://zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf. [12] Jones B. W., Pall G. (1939) Regular and semi-regular positive ternary quadratic forms. Acta Math. 70: 165-191. [13] Ju J., Oh B.-K., Seo B. (2019) Ternary universal sums of generalized polygonal numbers. Int. J. Number Theory 15: 655-675. [14] Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Math., Vol. 106, Cambridge, 1993. 10.1017/CBO9780511666155 1245266 [15] Oh B.-K. (2011) Ternary universal sums of generalized pentagonal numbers. J. Korean Math. Soc. 48: 837-847. [16] Oh B.-K., Sun Z.-W. (2009) Mixed sums of squares and triangular numbers (Ⅲ). J. Number Theory 129: 964-969. [17] O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York, 1963. 0152507 [18] Ramanujan S. (1917) On the expression of a number in the form $ax^2+by^2+cz^2+dw^2$. Proc. Cambridge Philos. Soc. 19: 11-21. [19] Sun Z.-W. (2007) Mixed sums of squares and triangular numbers. Acta Arith. 127: 103-113. [20] Sun Z.-W. (2015) On universal sums of polygonal numbers. Sci. China Math. 58: 1367-1396. [21] Sun Z.-W. (2016) A result similar to Lagrange's theorem. J. Number Theory 162: 190-211. [22] Sun Z.-W. (2017) On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$. J. Number Theory 171: 275-283. [23] Z.-W. Sun, Sequence A286944 in OEIS, 2017., Available from: http://oeis.org/A286944. [24] Z.-W. Sun, Universal sums of three quadratic polynomials, Sci. China Math., 2018. Available from: https://doi.org/10.1007/s11425-017-9354-4. See also arXiv: 1502.03056. 10.1007/s11425-017-9354-4 [25] Sun Z.-W. (2018) On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$. Nanjing Univ. J. Math. Biquarterly 35: 85-199. [26] Yang T. (1998) An explicit formula for local densities of quadratic forms. J. Number Theory 72: 309-356.
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