### 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 $x^2+y^2+32z^2$. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 1.833

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