Even universal sums of triangular numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917754794147840 |
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| author | Ju, Jangwon |
| author_facet | Ju, Jangwon |
| contents | For an arbitrary integer $x$, an integer of the form $T(x)\!=\!\frac{x^2+x}{2}$ is called a triangular number. Let $α_1,\dots,α_k$ be positive integers. A sum $Δ_{α_1,\dots,α_k}(x_1,\dots,x_k)=α_1 T(x_1)+\cdots+α_k T(x_k)$ of triangular numbers is said to be even universal if the Diophantine equation $Δ_{α_1,\dots,α_k}(x_1,\dots,x_k)=2n$ has an integer solution $(x_1,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$. In this article, we classify all even universal sums of triangular numbers. Furthermore, we provide an effective criterion on even universality of an arbitrary sum of triangular numbers, which is a generalization of the triangular theorem of eight of Bosma and Kane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Even universal sums of triangular numbers Ju, Jangwon Number Theory 11E12, 11E20 For an arbitrary integer $x$, an integer of the form $T(x)\!=\!\frac{x^2+x}{2}$ is called a triangular number. Let $α_1,\dots,α_k$ be positive integers. A sum $Δ_{α_1,\dots,α_k}(x_1,\dots,x_k)=α_1 T(x_1)+\cdots+α_k T(x_k)$ of triangular numbers is said to be even universal if the Diophantine equation $Δ_{α_1,\dots,α_k}(x_1,\dots,x_k)=2n$ has an integer solution $(x_1,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$. In this article, we classify all even universal sums of triangular numbers. Furthermore, we provide an effective criterion on even universality of an arbitrary sum of triangular numbers, which is a generalization of the triangular theorem of eight of Bosma and Kane. |
| title | Even universal sums of triangular numbers |
| topic | Number Theory 11E12, 11E20 |
| url | https://arxiv.org/abs/2408.11310 |