Improved Bounds for the Index Conjecture in Zero-Sum Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912645852954624 |
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| author | Pendleton, Andrew |
| author_facet | Pendleton, Andrew |
| contents | The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for $n>10^{20}$. In this paper, we prove that said upper bound can be reduced to $4.6\cdot10^{13}$, and lower under certain coprimality conditions. Further, we verify the conjecture for $n<1.8\cdot10^6$ through the application of High Performance Computing (HPC). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds for the Index Conjecture in Zero-Sum Theory Pendleton, Andrew Number Theory Combinatorics The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for $n>10^{20}$. In this paper, we prove that said upper bound can be reduced to $4.6\cdot10^{13}$, and lower under certain coprimality conditions. Further, we verify the conjecture for $n<1.8\cdot10^6$ through the application of High Performance Computing (HPC). |
| title | Improved Bounds for the Index Conjecture in Zero-Sum Theory |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2510.11976 |