Square-weighted zero-sum constants
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913302864461824 |
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| author | Paul, Krishnendu Paul, Shameek |
| author_facet | Paul, Krishnendu Paul, Shameek |
| contents | Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a non-zero square in $\mathbb Z_n$. We determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence. We also determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence whose terms are consecutive terms of the given sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_13143 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Square-weighted zero-sum constants Paul, Krishnendu Paul, Shameek Number Theory 11B50 (primary), 11B75 (secondary) Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a non-zero square in $\mathbb Z_n$. We determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence. We also determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence whose terms are consecutive terms of the given sequence. |
| title | Square-weighted zero-sum constants |
| topic | Number Theory 11B50 (primary), 11B75 (secondary) |
| url | https://arxiv.org/abs/2202.13143 |