On a certain arithmetic function defined via Bernoulli numbers

Technical Summary: On a Certain Arithmetic Function Defined via Bernoulli Numbers

Problem Statement
The paper investigates the integrality properties of a specific arithmetic function, $\chi(n)$, defined on the set of odd integers $n \ge 3$. This function is constructed using Bernoulli numbers ($B_j$) and aims to establish a unified criterion that distinguishes between three distinct classes of numbers: prime numbers, Carmichael numbers, and Giuga numbers. The central problem is to determine for which odd integers $n$ the value $\chi(n)$ (or its scaled variant $\tilde{\chi}(n)$) yields an integer.

Methodology
The author employs elementary number theory, relying heavily on the classical von Staudt-Clausen theorem regarding the denominators of Bernoulli numbers. The methodology proceeds as follows:

1. Definition of the Function: The function is defined as:
   $$\chi(n) = \frac{\text{denom}\left(\frac{B_{n-1}}{n-1}\right)}{n \cdot \text{denom}\left(B_{n-1} + \frac{1}{n}\right)}$$
   where $B_{n-1} = A/D$ is the Bernoulli number in reduced form. The paper also considers the scaled function $\tilde{\chi}(n) = n \cdot \chi(n)$.

2. Application of von Staudt-Clausen: The analysis utilizes the theorem stating that for any positive odd integer $n$, $B_{n-1} + \sum \frac{1}{p}$ is an integer, where the sum runs over primes $p$ such that $(p-1) \mid (n-1)$. This allows for the precise determination of the denominators involved in $\chi(n)$.

3. Case Analysis: The paper systematically evaluates $\chi(n)$ for three cases:

   • Prime numbers ($p$): Using the properties of the denominator of $B_{p-1}$, the author proves $\chi(p)$ is always an integer.
   • Composite non-Carmichael numbers: The author demonstrates that if $n$ is composite and fails Korselt's criterion (i.e., there exists a prime divisor $q$ such that $(q-1) \nmid (n-1)$), then $\chi(n)$ is not an integer.
   • Carmichael numbers: For Carmichael numbers (square-free composites where $(p-1) \mid (n-1)$ for all prime divisors $p$), the author simplifies the expression for $\tilde{\chi}(n)$ and shows it is always an integer.

4. Connection to Giuga Numbers: The paper further refines the condition for Carmichael numbers. It establishes that $\tilde{\chi}(n)$ is divisible by $n$ (i.e., $n \mid \tilde{\chi}(n)$) if and only if $n$ satisfies the condition for being a Giuga number ($n \equiv p \pmod{p^2}$ for all prime divisors $p$).

Key Contributions and Results
The primary result of the paper is Theorem 1.6, which provides a necessary and sufficient condition for the integrality of $\chi(n)$:

The function $\chi(n)$ is integer-valued if and only if $n$ is either a prime number or an odd Giuga number.

This result effectively unifies the characterization of primes and the hypothetical odd Giuga numbers under a single integrality criterion derived from Bernoulli numbers.

Additional findings include:

• Lemma 1.2: For any prime $p$, $\chi(p)$ is an even integer.
• Empirical Classification: The paper observes that the condition $\chi(p) = \lambda$ partitions the set of prime numbers into disjoint classes $P_\lambda$. The author provides a table of the first ten primes for various even values of $\lambda$ (ranging from 2 to 100).
• Conjecture 2.1: The paper proposes a conjecture linking the divisibility of the numerator of $B_{p-1}$ by $(p-1)/2$ to the representability of the prime $p$ by the quadratic form $-x^2 + 2xy + 2y^2$.

Significance and Claims
The paper claims to offer an "elementary" approach accessible to a broad audience that unifies three distinct number-theoretic concepts. By constructing $\chi(n)$, the author demonstrates that the property of being an integer serves as a filter:

• If $\chi(n) \in \mathbb{Z}$, then $n$ is either prime or an odd Giuga number.
• If $n$ is a Carmichael number that is not a Giuga number, $\chi(n)$ is not an integer (though $\tilde{\chi}(n)$ is).

The author notes that the existence of odd Giuga numbers remains an open problem (with known lower bounds suggesting they would have at least 19,908 decimal digits if they exist). Consequently, the paper does not claim to prove the existence of odd Giuga numbers but rather provides a theoretical framework where their existence would be equivalent to the existence of composite odd integers $n$ for which $\chi(n)$ is an integer. The work is presented as a state assignment project of the Ministry of Education and Science of the Russian Federation.