发表机构
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences（俄罗斯科学院西伯利亚分院系统动力学与控制理论马特罗佐夫研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究定义在奇数\(n\geq3\)上、通过伯努利数构造的算术函数，该函数将素数、卡迈克尔数和久加数统一到一个整除性准则中，为相关研究提供新视角。
AI 中文摘要
我们研究了一个定义在奇数\(n\geq3\)集合上的算术函数的整除性性质。它是通过伯努利数构造的。结果表明，这个函数将素数、卡迈克尔数和久加数这三个不同的数类统一到一个单一的整除性准则中。本文内容基础，广大读者均可理解。
英文摘要
The paper defines a certain function \(χ(n)\) on the set of odd integers \(n \ge 3\). The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if \(n\) is a prime number or an odd Giuga number. Thus, the integrality of \(χ(n)\) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt--Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers \(p\) for which \(χ(p)=p-1\) and Novák--Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes \(P_μ\) of primes with a fixed value \(χ(p)=μ\), a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number \(B_{p-1}\). For some \(μ\), explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.
Comments9 pages
