Arithmetic progressions represented by diagonal ternary quadratic forms
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916086976348160 |
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| author | Wu, Hai-Liang Sun, Zhi-Wei |
| author_facet | Wu, Hai-Liang Sun, Zhi-Wei |
| contents | Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and $3x^2+5y^2+15z^2$ is $(15,8)$-universal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1811_05855 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Arithmetic progressions represented by diagonal ternary quadratic forms Wu, Hai-Liang Sun, Zhi-Wei Number Theory 11E25, 11B25, 11E20 Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and $3x^2+5y^2+15z^2$ is $(15,8)$-universal. |
| title | Arithmetic progressions represented by diagonal ternary quadratic forms |
| topic | Number Theory 11E25, 11B25, 11E20 |
| url | https://arxiv.org/abs/1811.05855 |