Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917050684801024 |
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| author | Kellner, Bernd C. |
| author_facet | Kellner, Bernd C. |
| contents | Extending previous work of the author, we compute the Wilson quotient modulo $p^5$ and $p^6$, and equivalently $(p-1)!$ modulo $p^6$ and $p^7$, respectively. Further, we determine some power sums of the Fermat quotients up to modulo $p^6$. Subsequently, we discuss some patterns that occur in the $p$-adic coefficients of the Wilson quotient as well as of $(p-1)!$, whereby the original congruence $(p-1)! \equiv -1 \pmod{p}$ fits perfectly into the theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$ Kellner, Bernd C. Number Theory 11B65, 11B68 Extending previous work of the author, we compute the Wilson quotient modulo $p^5$ and $p^6$, and equivalently $(p-1)!$ modulo $p^6$ and $p^7$, respectively. Further, we determine some power sums of the Fermat quotients up to modulo $p^6$. Subsequently, we discuss some patterns that occur in the $p$-adic coefficients of the Wilson quotient as well as of $(p-1)!$, whereby the original congruence $(p-1)! \equiv -1 \pmod{p}$ fits perfectly into the theory. |
| title | Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$ |
| topic | Number Theory 11B65, 11B68 |
| url | https://arxiv.org/abs/2510.26743 |