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 $K_p \equiv \mathbf{Bell}_{p-1} - 1 \pmod p$, 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 $p$ is resides in the poor man's adele ring $\mathcal{A}$ .

• The paper develops Kurepa–Bell–Wilson congruences connecting left factorials, Bell numbers, Wilson quotients, and related Fermat and Agoh–Giuga quotients.
• The Gertsch quotient is shown to be integral, while its congruence with the Wilson quotient occurs only at primes 3, 7, and 2887 below 3000, separating a rare coincidence from a general law.
• The paper packages these quantities in Kontsevich’s poor man’s adele ring and reformulates Kurepa’s conjecture as the non-vanishing of an associated adele element, without resolving the conjecture.

This paper by Francis Atta Howard studies the interplay between Kurepa's left factorial conjecture, Bell numbers modulo primes, and classical quotient congruences (Wilson, Fermat, Lerch, Agoh–Giuga), culminating in a "Kurepa–Bell–Wilson" (KBW) congruence and an interpretation of the resulting Gertsch quotient as an element of Kontsevich's "poor man's adele ring" $$\mathcal{A} = \left(\prod_p \mathbb{Z}/p\mathbb{Z}\right)/\left(\bigoplus_p \mathbb{Z}/p\mathbb{Z}\right)$$ (2604.09714).

Background: Kurepa, Gertsch, and Bell modulo p

The left factorial $!n = \sum_{m=0}^{n-1} m!$ was introduced by Kurepa in 1971 with the conjecture that $\gcd(!n, n!) = 2$ for $n > 2$, equivalently that $K_p := !p \not\equiv 0 \pmod{p}$ for every odd prime $p > 3$. The paper builds on Gertsch's thesis result that

$$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$

which follows from Wilson's theorem applied to the final summand $(p-1)!$, together with Touchard-type congruences for Bell polynomials and the identity $\mathbf{Der}_{p-1} \equiv K_p \pmod{p}$ relating derangement numbers to left factorials in $\mathbf{GF}(p)$. These equivalences underpin the reformulations of the Kurepa conjecture pursued by Barsky, Benzaghou, Mijajlović, Sun, Zagier, and Vladimirov.

The KBW congruence and related quotient identities

The central observation is that dividing the Kurepa congruence by $!n = \sum_{m=0}^{n-1} m!$0 produces a family of congruences at the level of quotients. Writing $!n = \sum_{m=0}^{n-1} m!$1 for the Wilson quotient, the paper establishes:

$!n = \sum_{m=0}^{n-1} m!$2

with parallel forms via Lerch's formula $!n = \sum_{m=0}^{n-1} m!$3 (KBL), and, specializing the Fermat quotient $!n = \sum_{m=0}^{n-1} m!$4 at $!n = \sum_{m=0}^{n-1} m!$5 and $!n = \sum_{m=0}^{n-1} m!$6, Wieferich and Mirimanoff variants (KBF, KBM). Analogous identities are derived using the Agoh–Giuga quotient $!n = \sum_{m=0}^{n-1} m!$7, yielding $!n = \sum_{m=0}^{n-1} m!$8, where $!n = \sum_{m=0}^{n-1} m!$9 is Agoh's special quotient. In the adele ring these become identities among elements such as $\gcd(!n, n!) = 2$0 and $\gcd(!n, n!) = 2$1, where $\gcd(!n, n!) = 2$2 is Kaneko's finite logarithm satisfying $\gcd(!n, n!) = 2$3.

The Gertsch quotient

The Gertsch quotient is defined as

$\gcd(!n, n!) = 2$4

which the author proves is always an integer. A key computational finding is that $\gcd(!n, n!) = 2$5 holds only for $\gcd(!n, n!) = 2$6 among primes up to 3000 — a sparse coincidence rather than a general law, a point the paper states plainly. Notably, at the known Wilson primes $\gcd(!n, n!) = 2$7, the Gertsch quotient does not vanish modulo $\gcd(!n, n!) = 2$8, so vanishing of $\gcd(!n, n!) = 2$9 does not force vanishing of $n > 2$0. No prime with $n > 2$1 ("Gertsch prime") is known; the paper shows that if a Gertsch prime existed with $n > 2$2 and $n > 2$3, it would also be a Wilson prime, and conversely that absence of Gertsch primes implies the Kurepa conjecture. This conditional structure means the results do not prove Kurepa's conjecture but recast it in terms of the non-vanishing of $n > 2$4.

Vladimirov atoms and the Wilson quotient

The paper revisits Vladimirov's 2002 criterion linking left factorials and Bernoulli numbers,

$n > 2$5

and rewrites it entirely in terms of the Wilson quotient using Agoh's congruences ($n > 2$6, $n > 2$7). The resulting analogue states that Kurepa's conjecture is equivalent to the non-vanishing of $n > 2$8 modulo $n > 2$9, given $K_p := !p \not\equiv 0 \pmod{p}$0. Numerically (via PARI/GP checks attributed to Kaneko), $K_p := !p \not\equiv 0 \pmod{p}$1 occurs at $K_p := !p \not\equiv 0 \pmod{p}$2 and $K_p := !p \not\equiv 0 \pmod{p}$3 at $K_p := !p \not\equiv 0 \pmod{p}$4, so these atoms do vanish at isolated primes — meaning the sufficient conditions in Vladimirov's theorem are not vacuous but also not automatic.

Extensions: Hodge integrals and finite Euler constants

Two further extensions are developed. First, mimicking Vladimirov's theorem with the Faber–Pandhariponde Hodge integral constants $K_p := !p \not\equiv 0 \pmod{p}$5, the author asks whether $K_p := !p \not\equiv 0 \pmod{p}$6 bears on Kurepa's conjecture; Don Zagier, in a private communication cited in the paper, judged this approach unlikely to yield information about $K_p := !p \not\equiv 0 \pmod{p}$7, and the paper reports this skepticism honestly. Second, following Kaneko–Matsusaka–Seki's finite analogues of Euler's constant, the author defines elements $K_p := !p \not\equiv 0 \pmod{p}$8, $K_p := !p \not\equiv 0 \pmod{p}$9, and $p > 3$0 of $p > 3$1 and derives identities connecting them to $p > 3$2 and $p > 3$3, e.g., $p > 3$4 and its corollaries involving Gregory coefficients.

The adele ring argument

The main structural claim (Theorem on the Kaneko ring) is that $p > 3$5 lies in $p > 3$6 and is nonzero. The argument combines the KBW congruence with the empirical fact that $p > 3$7 never vanishes simultaneously with $p > 3$8 at $p > 3$9, concluding that $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$0 cannot vanish. It should be noted that this non-vanishing argument rests on computational evidence over a finite range plus the assumption that no larger coincidences occur; it is a heuristic transfer of the Kurepa conjecture into the adele-theoretic framework rather than a proof, consistent with the paper's own framing of $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$1 via the absence of known Gertsch primes.

Limitations and open questions

The paper concedes several open points. The equivalence between Gertsch-prime nonexistence and the Kurepa conjecture is conditional; verifying it requires searching for Wilson primes beyond $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$2 (per Crandall–Dilcher–Pomerance and Costa–Gerbicz–Harvey search bounds), which remains computationally out of reach. Whether infinitely many primes make $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$3, $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$4, $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$5, and $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$6 simultaneously integral (only $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$7 is observed, with $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$8) is posed as a question. The proposed Bernoulli-analogue congruence $$K_p = \sum_{n=0}^{p-1} n! \equiv \mathbf{Bell}_{p-1} - 1 \pmod{p},$$9 (KBB) is offered as a target for computation rather than established as restrictive. Finally, whether $(p-1)!$0 itself is zero in $(p-1)!$1 — a question Kaneko et al. explicitly leave open — remains unresolved here.

Conclusion

The paper organizes a web of congruences connecting Kurepa factorials, Bell numbers, derangements, Bernoulli numbers, and the classical Fermat/Wilson/Lerch/Agoh–Giuga quotients, and packages the resulting Gertsch quotient as a well-defined element of the poor man's adele ring. Its principal contributions are the KBW/KBL/KBF congruence family, the identification of the rare coincidences $(p-1)!$2 at $(p-1)!$3, and a reformulation of the Kurepa conjecture as the non-vanishing of $(p-1)!$4 in $(p-1)!$5. The reformulation is elegant and computationally supported within the checked range, but the underlying arithmetic questions — existence of Gertsch or additional Wilson primes, and the vanishing of $(p-1)!$6 in $(p-1)!$7 — remain open.