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

This paper investigates a Kurepa-Bell-Wilson congruence involving left factorials and Bell numbers to demonstrate that it naturally generates the non-zero Gertsch quotient, which resides within the "poor man's adele ring" for larger primes.

Original authors: Francis Atta Howard

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francis Atta Howard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery that has puzzled mathematicians for over 50 years. The mystery is called the Kurepa Conjecture.

In simple terms, this conjecture asks a very specific question about a special kind of number sum called a "Left Factorial." If you add up factorials (like $0! + 1! + 2! + \dots$ ) up to a certain point, does the total ever divide perfectly by a prime number (like 3, 5, 7, 11)? The conjecture says no, it never divides perfectly.

This paper, written by Francis Atta Howard, is a report on a new investigation into this mystery. The author doesn't just look at the numbers; he builds a whole new "map" to navigate them. Here is the breakdown of the paper using everyday analogies:

1. The Main Characters: The "Left Factorial" and the "Bell"

Think of the Left Factorial ( $K_p$ ) as a giant pile of bricks. You stack them up one by one.

The Mystery: Does this pile of bricks ever fit perfectly into a box of size $p$ (where $p$ is a prime number)? The Kurepa conjecture says: "No, there's always a little bit of brick sticking out."

The author introduces a second character: Bell Numbers. Think of these as a different way of organizing the same bricks. The paper highlights a famous discovery: The pile of Left Factorial bricks is almost exactly the same as the Bell Number pile, minus just one tiny brick.

Analogy: If the Bell pile has 100 bricks, the Left Factorial pile has 99. They are twins separated by a single unit.

2. The New Tool: The "Poor Man's Adele Ring"

This sounds like a fancy, expensive building, but the author calls it the "Poor Man's Adele Ring."

The Metaphor: Imagine you have a giant library where every book represents a different prime number (Book 3, Book 5, Book 7, etc.). Usually, you have to look at each book individually.
The "Poor Man's Adele Ring" is like a super-index or a master key that lets you look at all the books at once. It allows the author to see patterns that connect all these different prime numbers together, rather than checking them one by one. It's a "cheap" (hence "poor man's") but powerful way to organize infinite amounts of data.

3. The "Gertsch Quotient": The New Detective

The author creates a new mathematical tool called the Gertsch Quotient ( $G_p$ ).

The Metaphor: Imagine you have two suspects (the Left Factorial and the Bell Number). You know they are almost identical, but you need to know exactly how they differ. The Gertsch Quotient is the magnifying glass that measures the tiny difference between them.
The paper proves that this "magnifying glass" always gives you a whole number (an integer).
The Big Claim: If the Kurepa Conjecture is false (meaning the Left Factorial does divide perfectly by a prime), then this "magnifying glass" would have to show a result of zero. But the author shows that, so far, the glass never shows zero.

4. Connecting the Dots: The "Congruence"

The paper is full of equations that look like magic spells, but they are actually just connecting different famous mathematical ideas:

Wilson's Theorem: A rule about how factorials behave with primes.
Fermat's Little Theorem: Another rule about powers and primes.
The Connection: The author shows that the "Left Factorial" mystery is linked to these other rules. He creates a bridge (a "congruence") that says:

"The difference between the Left Factorial and the Bell Number is related to the Wilson Quotient (another famous number) and some other constants."

It's like finding out that the mystery of the missing brick is actually the same mystery as the missing piece in a different puzzle you've been solving for years.

5. The "Atoms" and "Genus"

The paper gets a bit more abstract in the middle, talking about "Vladimirov atoms" and "Genus."

The Metaphor: Think of the numbers as being made of tiny, invisible atoms. The author is trying to see if these atoms vibrate in a specific way that would prove the Kurepa conjecture. He also looks at "Genus," which in math is like the number of holes in a donut. He asks: "Does the shape of the donut (the math structure) tell us anything about our brick pile?"
The Result: He finds some interesting links, but admits that some of these connections (like the ones involving "Hodge integrals") might be too complex to solve the mystery directly yet.

6. The Conclusion: "We Haven't Found a Counter-Example"

The paper ends with a lot of computer checks (numerical observations).

The author and his colleagues checked millions of prime numbers.
They looked for a "Gertsch Prime"—a prime number where the Gertsch Quotient equals zero.
The Result: They found none.
The Takeaway: Every time they checked, the "magnifying glass" showed a non-zero number. This supports the idea that the Kurepa Conjecture is true: The Left Factorial never divides perfectly by a prime.

Summary for the General Audience

This paper is a detective story in the world of numbers.

The Crime: A mathematician named Kurepa claimed that a specific sum of numbers never divides evenly by a prime.
The Investigation: Francis Atta Howard built a new "map" (the Poor Man's Adele Ring) to look at all primes at once.
The Clue: He created a new measuring tool (the Gertsch Quotient) that links the crime to other famous mathematical rules (Wilson, Fermat, Bell).
The Verdict: After checking millions of numbers, the tool never gave a "zero" reading. This strongly suggests the original claim is true, even though a final, perfect mathematical proof is still being sought.

The paper is a mix of deep theory (building new bridges between math concepts) and hard work (checking millions of numbers on a computer) to keep the mystery of the "Left Factorial" alive.

1. Problem Statement

The paper addresses the Kurepa Conjecture, proposed by Duro Kurepa in 1971. The conjecture states that for any integer $n > 2$ , the greatest common divisor of the left factorial $!n$ and $n!$ is 2. Equivalently, for every odd prime $p > 3$ , the left factorial $!p = \sum_{k=0}^{p-1} k!$ is not congruent to 0 modulo $p$ (i.e., $!p \not\equiv 0 \pmod p$ ).

Despite extensive computational verification and various theoretical approaches involving Bernoulli numbers, derangements, and Galois fields, the conjecture remains unproven. This paper seeks to:

Establish a new congruence linking Kurepa factorials, Bell numbers, and Wilson quotients.
Define a new mathematical object, the Gertsch Quotient ( $G_p$ ).
Embed these congruences into the "poor man's adele ring" ( $A$ ), a structure defined by Kaneko and Zagier involving finite multiple zeta values.
Investigate whether the non-vanishing of $G_p$ modulo $p$ provides a pathway to proving or disproving the Kurepa conjecture.

2. Methodology

The author employs a blend of number-theoretic analysis, algebraic manipulation, and heuristic arguments based on $p$ -adic analysis. Key methodological components include:

Congruence Derivation: The paper derives relationships between the Kurepa factorial ( $K_p$ ), Bell numbers ( $Bell_{p-1}$ ), and Wilson's theorem. It utilizes known identities such as $K_p \equiv Bell_{p-1} - 1 \pmod p$ and $(p-1)! \equiv -1 \pmod p$ .
Definition of New Quotients: The author defines the Gertsch Quotient as:
$G_p = \frac{(K_p - Bell_{p-1}) + 1}{p}$
This quotient is shown to be an integer for all odd primes $p \ge 3$ .
Algebraic Embedding: The study embeds these modular quotients into the ring $A$ , defined as the quotient of the direct product of $\mathbb{Z}/p\mathbb{Z}$ by the direct sum ideal. This ring, referred to as the "poor man's adele ring," allows for the treatment of sequences of modular residues as single algebraic elements.
Analogous Structures: The paper draws parallels between the Gertsch quotient and other famous number-theoretic quotients, including the Wilson quotient ( $W_p$ ), Lerch quotient ( $L_p$ ), Fermat quotient ( $q_p$ ), and Agoh-Giuga quotient ( $AG_p$ ).
Connection to Special Functions: The methodology extends to connections with Vladimirov atoms (involving Bernoulli numbers), Hodge integrals (via Faber-Pandhariponde constants $b_g$ ), and Euler's constant analogues ( $\gamma_A$ ).
Computational Verification: The author uses software (PARI/GP) to verify congruences for primes up to $10^6$ and checks specific values for known Wilson primes and Gertsch primes.

3. Key Contributions

A. The Kurepa-Bell-Wilson (KBW) Congruence

The paper establishes a fundamental congruence linking three major concepts:
$\frac{K_p + 1}{p} \equiv \frac{Bell_{p-1}}{p} + W_p \pmod p$
where $W_p = \frac{(p-1)! + 1}{p}$ is the Wilson quotient. This identity unifies the study of left factorials with Bell numbers and Wilson's theorem.

B. The Gertsch Quotient ( $G_p$ )

The author introduces $G_p$ as a critical invariant. The paper proves that:

$G_p$ is always an integer.
$G_p \equiv W_p \pmod p$ for specific primes (e.g., $p=3, 7, 2887$ ).
Crucially, the Kurepa conjecture ( $K_p \not\equiv 0 \pmod p$ ) is equivalent to the statement that $G_p \not\equiv 0 \pmod p$ for all odd primes $p \ge 3$ . If a prime exists where $G_p \equiv 0 \pmod p$ , the Kurepa conjecture would be false.

C. Embedding in the "Poor Man's Adele Ring" ( $A$ )

The paper defines elements in the ring $A$ corresponding to these quotients:

$\gamma^W_A = (W_p \pmod p)_p$
$\gamma^G_A = (G_p \pmod p)_p$
$\gamma^{K_p}_A = (K_p \pmod p)_p$
The study demonstrates that $\gamma^{K_p}_A$ resides in $A$ and is non-zero, provided no "Gertsch primes" (primes where $G_p \equiv 0 \pmod p$ ) exist.

D. Connections to Other Constants

Vladimirov Atoms: The paper revisits Vladimirov's 2002 equivalent of the Kurepa conjecture involving Bernoulli numbers and extends it to relate to the Wilson quotient.
Euler Constant Analogues: It defines analogues of Euler's constant ( $\gamma$ ) within the ring $A$ using the Gertsch, Lerch, and Agoh-Giuga quotients, showing relationships like $\gamma^G_A + \log_A(m) = \gamma^Q_A(m)$ .
Hodge Integrals: The author explores a potential link between the Kurepa factorial and Faber-Pandhariponde constants ( $b_g$ ), though this connection is noted as heuristic and potentially "unnatural" based on feedback from Don Zagier.

4. Results

Non-Vanishing of $G_p$ : Extensive numerical checks (up to $p \approx 61$ and beyond via literature) show that $G_p \not\equiv 0 \pmod p$ for all tested primes. No "Gertsch primes" have been found.
Relation to Wilson Primes: The paper notes that while Wilson primes ( $W_p \equiv 0 \pmod p$ ) are known ( $p=5, 13, 563$ ), these do not necessarily imply $G_p \equiv 0 \pmod p$ . In fact, for $p=5, 13, 563$ , $G_p$ remains non-zero modulo $p$ .
Theorem 1 & 2:
- Theorem 1: Confirms that the element $\gamma^{K_p}_A$ in the adele ring $A$ is non-zero, assuming the non-existence of Gertsch primes.
- Theorem 2: Provides the explicit KBW, KBL (Kurepa-Bell-Lerch), KBF (Kurepa-Bell-Fermat), and KBM (Kurepa-Bell-Mirimanoff) congruences.
Corollary 1: The Kurepa conjecture holds true if and only if there are no odd primes $p \ge 3$ such that $G_p \equiv 0 \pmod p$ .
Numerical Data: The paper provides tables of values for $G_p$ , $AG_p$ , $W_p$ , and $Bell_{p-1}$ for various primes, illustrating the behavior of these sequences.

5. Significance

New Perspective on Kurepa: By reframing the Kurepa conjecture in terms of the Gertsch quotient and its relationship to the Wilson quotient, the paper offers a fresh algebraic approach to a long-standing open problem.
Unification of Concepts: The work successfully unifies disparate areas of number theory: left factorials, Bell numbers, derangements, Wilson's theorem, Bernoulli numbers, and finite multiple zeta values.
Algebraic Framework: The use of the "poor man's adele ring" provides a sophisticated algebraic setting to analyze sequences of modular residues, potentially opening new avenues for proving congruences that are difficult to handle with standard modular arithmetic alone.
Heuristic Guidance: While the paper does not provide a rigorous proof of the Kurepa conjecture, it establishes strong heuristic evidence (via the non-vanishing of $G_p$ ) and identifies specific conditions (the non-existence of Gertsch primes) that must be met for the conjecture to hold.

In conclusion, the paper proposes that the Kurepa conjecture is deeply intertwined with the properties of the Gertsch quotient within the adele ring $A$ . The non-existence of a prime $p$ where $G_p \equiv 0 \pmod p$ is presented as the necessary and sufficient condition for the conjecture to be true.