Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$
Bernd C. Kellner
TL;DR
This work extends Wilson-quotient congruences to higher prime powers by linking the Wilson quotient $\\mathcal{W}_p = \frac{(p-1)!+1}{p}$ to power sums of Fermat quotients via a polynomial transform. It introduces and utilizes divided Bernoulli numbers $\\overline{\\mathcal{B}}_n$ and generalized Kummer congruences to express $\\mathcal{W}_p$ modulo $p^5$ and $p^6$ (and $(p-1)!$ modulo $p^6$ and $p^7$) in terms of explicit coefficients $\\omega_\\nu$. The main contributions are the explicit constructions of the polynomials $\\psi_\\nu$ and their tilde versions, the derivation of closed forms for $\\omega_1$ through $\\omega_6$ in terms of $\\overline{\\mathcal{B}}_n$ and related numbers, and the systematic organization of the reduction process via $Q_p(n)$ congruences. The results illuminate the $p$-adic structure of Wilson-type congruences, reveal patterns among the coefficients, and highlight computational challenges that grow with the modulus, offering a platform for further exploration of supercongruences in this setting.
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.