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

Francis Atta Howard

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arXiv · 2604.09714

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

Francis Atta Howard

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Abstract

Wilson's theorem is notably related to left factorials, expressed as $K_p \equiv \mathbf{Bell}_{p-1} - 1 \pmod p$, for prime $p\geq3$. This study examines a Kurepa-Bell-Wilson congruence (\textbf{KBW}), $\frac{K_p + 1}{p}\equiv \frac{ \mathbf{Bell}_{p-1}}{p}+ W_p \pmod{p}$, 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 $\mathcal{A}$ .

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Francis Atta Howard. 2026-04-21. Gertsch quotient living in the "poor man's adele ring" $\mathcal{A}$: Kurepa-Bell-Wilson congruence. https://arxiv.org/abs/2604.09714

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