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

Some universal quadratic sums over the integers

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Zhi-Wei Sun

Received  August 2019 Published  December 2019

Fund Project: The initial version of this paper was posted to arXiv (with the code arXiv:1707.06223) in July 2017. The second author is supported by the National Natural Science Foundation of China (grant 11571162) and the NSFC-RFBR Cooperation and Exchange Program (grant 11811530072)

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. Electronic Research Archive, 2019, 27: 69-87. doi: 10.3934/era.2019010
References:
[1]

B. C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/stml/034.  Google Scholar

[2] J. W. S. Cassels, Rational Quadratic Forms, Academic Press, London, 1978.   Google Scholar
[3]

L. E. Dickson, Quaternary quadratic forms representing all integers, Amer. J. Math., 49 (1927), 39-56.  doi: 10.2307/2370770.  Google Scholar

[4] L. E. Dickson, Modern Elementary Theory of Numbers, Univ. of Chicago Press, Chicago, 1939.   Google Scholar
[5]

A. G. Earnest, Congruence conditions on integers represented by ternary quadratic forms, Pacific J. Math., 90 (1980), 325-333.  doi: 10.2140/pjm.1980.90.325.  Google Scholar

[6]

A. G. Earnest, Representation of spinor exceptional integers by ternary quadratic forms, Nagoya Math. J., 93 (1984), 27-38.  doi: 10.1017/S0027763000020717.  Google Scholar

[7]

F. Ge and Z.-W. Sun, On some universal sums of generalized polygonals, Colloq. Math., 145 (2016), 149-155.   Google Scholar

[8]

S. Guo, H. Pan and Z.-W. Sun, Mixed sums of squares and triangular numbers (Ⅱ), Integers, 7 (2007), A56, 5pp (electronic).  Google Scholar

[9]

W. C. Jagy, Five regular or nearly-regular ternary quadratic forms, Acta Arith., 77 (1996), 361-367.  doi: 10.4064/aa-77-4-361-367.  Google Scholar

[10]

W. C. JagyI. Kaplansky and A. Schiemann, There are 913 regular ternary forms, Mathematika, 44 (1997), 332-341.  doi: 10.1112/S002557930001264X.  Google Scholar

[11]

W. C. Jagy, Integral Positive Ternary Quadratic Forms, Lecture Notes, 2014. Available from: http://zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf. Google Scholar

[12]

B. W. Jones and G. Pall, Regular and semi-regular positive ternary quadratic forms, Acta Math., 70 (1939), 165-191.  doi: 10.1007/BF02547347.  Google Scholar

[13]

J. JuB.-K. Oh and B. Seo, Ternary universal sums of generalized polygonal numbers, Int. J. Number Theory, 15 (2019), 655-675.  doi: 10.1142/S1793042119500350.  Google Scholar

[14]

Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Math., Vol. 106, Cambridge, 1993. doi: 10.1017/CBO9780511666155.  Google Scholar

[15]

B.-K. Oh, Ternary universal sums of generalized pentagonal numbers, J. Korean Math. Soc., 48 (2011), 837-847.  doi: 10.4134/JKMS.2011.48.4.837.  Google Scholar

[16]

B.-K. Oh and Z.-W. Sun, Mixed sums of squares and triangular numbers (Ⅲ), J. Number Theory, 129 (2009), 964-969.  doi: 10.1016/j.jnt.2008.10.002.  Google Scholar

[17]

O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York, 1963.  Google Scholar

[18]

S. Ramanujan, On the expression of a number in the form $ax^2+by^2+cz^2+dw^2$, Proc. Cambridge Philos. Soc., 19 (1917), 11-21.   Google Scholar

[19]

Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith., 127 (2007), 103-113.  doi: 10.4064/aa127-2-1.  Google Scholar

[20]

Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math., 58 (2015), 1367-1396.  doi: 10.1007/s11425-015-4994-4.  Google Scholar

[21]

Z.-W. Sun, A result similar to Lagrange's theorem, J. Number Theory, 162 (2016), 190-211.  doi: 10.1016/j.jnt.2015.10.014.  Google Scholar

[22]

Z.-W. Sun, On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$, J. Number Theory, 171 (2017), 275-283.  doi: 10.1016/j.jnt.2016.07.024.  Google Scholar

[23]

Z.-W. Sun, Sequence A286944 in OEIS, 2017., Available from: http://oeis.org/A286944. Google Scholar

[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. doi: 10.1007/s11425-017-9354-4.  Google Scholar

[25]

Z.-W. Sun, On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$, Nanjing Univ. J. Math. Biquarterly, 35 (2018), 85-199.   Google Scholar

[26]

T. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory, 72 (1998), 309-356.  doi: 10.1006/jnth.1998.2258.  Google Scholar

show all references

References:
[1]

B. C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/stml/034.  Google Scholar

[2] J. W. S. Cassels, Rational Quadratic Forms, Academic Press, London, 1978.   Google Scholar
[3]

L. E. Dickson, Quaternary quadratic forms representing all integers, Amer. J. Math., 49 (1927), 39-56.  doi: 10.2307/2370770.  Google Scholar

[4] L. E. Dickson, Modern Elementary Theory of Numbers, Univ. of Chicago Press, Chicago, 1939.   Google Scholar
[5]

A. G. Earnest, Congruence conditions on integers represented by ternary quadratic forms, Pacific J. Math., 90 (1980), 325-333.  doi: 10.2140/pjm.1980.90.325.  Google Scholar

[6]

A. G. Earnest, Representation of spinor exceptional integers by ternary quadratic forms, Nagoya Math. J., 93 (1984), 27-38.  doi: 10.1017/S0027763000020717.  Google Scholar

[7]

F. Ge and Z.-W. Sun, On some universal sums of generalized polygonals, Colloq. Math., 145 (2016), 149-155.   Google Scholar

[8]

S. Guo, H. Pan and Z.-W. Sun, Mixed sums of squares and triangular numbers (Ⅱ), Integers, 7 (2007), A56, 5pp (electronic).  Google Scholar

[9]

W. C. Jagy, Five regular or nearly-regular ternary quadratic forms, Acta Arith., 77 (1996), 361-367.  doi: 10.4064/aa-77-4-361-367.  Google Scholar

[10]

W. C. JagyI. Kaplansky and A. Schiemann, There are 913 regular ternary forms, Mathematika, 44 (1997), 332-341.  doi: 10.1112/S002557930001264X.  Google Scholar

[11]

W. C. Jagy, Integral Positive Ternary Quadratic Forms, Lecture Notes, 2014. Available from: http://zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf. Google Scholar

[12]

B. W. Jones and G. Pall, Regular and semi-regular positive ternary quadratic forms, Acta Math., 70 (1939), 165-191.  doi: 10.1007/BF02547347.  Google Scholar

[13]

J. JuB.-K. Oh and B. Seo, Ternary universal sums of generalized polygonal numbers, Int. J. Number Theory, 15 (2019), 655-675.  doi: 10.1142/S1793042119500350.  Google Scholar

[14]

Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Math., Vol. 106, Cambridge, 1993. doi: 10.1017/CBO9780511666155.  Google Scholar

[15]

B.-K. Oh, Ternary universal sums of generalized pentagonal numbers, J. Korean Math. Soc., 48 (2011), 837-847.  doi: 10.4134/JKMS.2011.48.4.837.  Google Scholar

[16]

B.-K. Oh and Z.-W. Sun, Mixed sums of squares and triangular numbers (Ⅲ), J. Number Theory, 129 (2009), 964-969.  doi: 10.1016/j.jnt.2008.10.002.  Google Scholar

[17]

O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York, 1963.  Google Scholar

[18]

S. Ramanujan, On the expression of a number in the form $ax^2+by^2+cz^2+dw^2$, Proc. Cambridge Philos. Soc., 19 (1917), 11-21.   Google Scholar

[19]

Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith., 127 (2007), 103-113.  doi: 10.4064/aa127-2-1.  Google Scholar

[20]

Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math., 58 (2015), 1367-1396.  doi: 10.1007/s11425-015-4994-4.  Google Scholar

[21]

Z.-W. Sun, A result similar to Lagrange's theorem, J. Number Theory, 162 (2016), 190-211.  doi: 10.1016/j.jnt.2015.10.014.  Google Scholar

[22]

Z.-W. Sun, On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$, J. Number Theory, 171 (2017), 275-283.  doi: 10.1016/j.jnt.2016.07.024.  Google Scholar

[23]

Z.-W. Sun, Sequence A286944 in OEIS, 2017., Available from: http://oeis.org/A286944. Google Scholar

[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. doi: 10.1007/s11425-017-9354-4.  Google Scholar

[25]

Z.-W. Sun, On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$, Nanjing Univ. J. Math. Biquarterly, 35 (2018), 85-199.   Google Scholar

[26]

T. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory, 72 (1998), 309-356.  doi: 10.1006/jnth.1998.2258.  Google Scholar

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