Some universal quadratic sums over the integers

Hai-Liang Wu; Zhi-Wei Sun; Hai-Liang Wu; Zhi-Wei Sun

doi:10.3934/era.2019010

Electronic Research Archive

2019, Volume 27, 69-87. doi: 10.3934/era.2019010

Previous Article Next Article

Some universal quadratic sums over the integers

Hai-Liang Wu ,
Zhi-Wei Sun ^,

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received: 01 August 2019
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 $.
- Universal sums,
- quadratic polynomials,
- ternary quadratic forms
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:

Abstract

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 $.

References

[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. London: 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. Chicago: 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.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)