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

TL;DR

This paper explores a new congruence involving Wilson's theorem, Bell numbers, and factorials, revealing that it naturally produces the Gertsch quotient within the structure of the poor man's adele ring.

Contribution

It introduces the Kurepa-Bell-Wilson congruence and shows its connection to the Gertsch quotient in the context of the adele ring, expanding understanding of these number theoretic objects.

Findings

The KBW congruence generates the Gertsch quotient for larger primes.

The Gertsch quotient resides in the poor man's adele ring $\

The study links classical number theory with the structure of the adele ring.

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 $\mathcal{A}$ .