Gertsch quotient living in the "poor man's adele ring" $\mathcal{A}$: Kurepa-Bell-Wilson congruence

Abstract

Wilson's theorem is notably related to left factorials, expressed as $K_p\equiv {\mathbf{Bell}}_{p-1}-1(\mathrm{mod}p)$, for prime $p\ge 3$. This study examines a Kurepa-Bell-Wilson congruence (\textbf{KBW}), $\frac{K_p+1}{p}\equiv \frac{{\mathbf{Bell}}_{p-1}}{p}+W_p(\mathrm{mod}p)$, and demonstrates that it naturally generates the non-zero "Gertsch quotient (${\mathbb{G}}_p$)," which, for larger primes modulo $p$ resides in the poor man's adele ring $\mathcal{A}$ .