On Graham's rearrangement conjecture
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918343299039232 |
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| author | Pham, Huy Tuan Sauermann, Lisa |
| author_facet | Pham, Huy Tuan Sauermann, Lisa |
| contents | Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq \mathbb{Z}_p\setminus \{0\}$ with $|S|\le p^{1-α}$ and $|S|$ sufficiently large with respect to $α$, for any $α\in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Graham's rearrangement conjecture Pham, Huy Tuan Sauermann, Lisa Combinatorics Discrete Mathematics Number Theory Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq \mathbb{Z}_p\setminus \{0\}$ with $|S|\le p^{1-α}$ and $|S|$ sufficiently large with respect to $α$, for any $α\in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$. |
| title | On Graham's rearrangement conjecture |
| topic | Combinatorics Discrete Mathematics Number Theory |
| url | https://arxiv.org/abs/2602.15797 |