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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.14605 |
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| _version_ | 1866915816303230976 |
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| author | Bulkhali, Nasser Abdo Saeed Sun, Zhi-Wei |
| author_facet | Bulkhali, Nasser Abdo Saeed Sun, Zhi-Wei |
| contents | An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal sums via products of Ramanujan's theta functions Bulkhali, Nasser Abdo Saeed Sun, Zhi-Wei Number Theory 11D72, 11E20, 11E25, 11F27, 14H42 An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal. |
| title | Universal sums via products of Ramanujan's theta functions |
| topic | Number Theory 11D72, 11E20, 11E25, 11F27, 14H42 |
| url | https://arxiv.org/abs/2410.14605 |