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

Wilson’s theorem is notably related to left factorials, expressed as $K_p\equiv {\mathbf{B}\mathbf{e}\mathbf{l}\mathbf{l}}_{p-1}-1(\mathrm{m}\mathrm{o}\mathrm{d}p)$, for prime $p\ge 3$. This study examines a Kurepa-Bell-Wilson congruence ($\mathbf{\text{KBW}}$), $\frac{K_p+1}{p}\equiv \frac{{\mathbf{B}\mathbf{e}\mathbf{l}\mathbf{l}}_{p-1}}{p}+W_p(\mathrm{m}\mathrm{o}\mathrm{d}p)$, 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 $\mathcal{A}$ .