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arXiv · 2510.26743

Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$

Bernd C. Kellner
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Abstract

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.

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Bernd C. Kellner. 2025-10-30. Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$. https://arxiv.org/abs/2510.26743

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