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

TL;DR

This paper extends the computation of Wilson quotients and factorials modulo higher prime powers, revealing patterns in p-adic coefficients and connecting to classical congruences.

Contribution

It advances the understanding of Wilson quotients modulo p^6 and p^7, and explores p-adic patterns in factorials and Fermat quotients.

Findings

Computed Wilson quotients modulo p^6 and p^7

Determined power sums of Fermat quotients up to p^6

Identified patterns in p-adic coefficients of Wilson quotients

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.