On the sequence $n! \bmod p$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909170000723968 |
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| author | Grebennikov, A. Sagdeev, A. Semchankau, A. Vasilevskii, A. |
| author_facet | Grebennikov, A. Sagdeev, A. Semchankau, A. Vasilevskii, A. |
| contents | We prove, that the sequence $1!, 2!, 3!, \dots$ produces at least $(\sqrt{2} + o(1))\sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $\mathcal{I} \subseteq \{0, 1, \dots, p - 1\}$ of length $N > p^{7/8 + \varepsilon}$ produce at least $(1 + o(1))\sqrt{p}$ distinct residues modulo $p$. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials $n_1! \dots n_7!$ modulo $p$, where $n_i = O(p^{6/7+\varepsilon})$ for all $i=1,\dots,7$, which provides a polynomial improvement upon the preceding results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_01153 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the sequence $n! \bmod p$ Grebennikov, A. Sagdeev, A. Semchankau, A. Vasilevskii, A. Number Theory Combinatorics 11N56, 11L03, 11B65 We prove, that the sequence $1!, 2!, 3!, \dots$ produces at least $(\sqrt{2} + o(1))\sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $\mathcal{I} \subseteq \{0, 1, \dots, p - 1\}$ of length $N > p^{7/8 + \varepsilon}$ produce at least $(1 + o(1))\sqrt{p}$ distinct residues modulo $p$. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials $n_1! \dots n_7!$ modulo $p$, where $n_i = O(p^{6/7+\varepsilon})$ for all $i=1,\dots,7$, which provides a polynomial improvement upon the preceding results. |
| title | On the sequence $n! \bmod p$ |
| topic | Number Theory Combinatorics 11N56, 11L03, 11B65 |
| url | https://arxiv.org/abs/2204.01153 |