arXiv 2103.048832021-03-05math.NT

关于广义Carmichael数

On Generalized Carmichael Numbers

Yongyi Chen, Tae Kyu Kim

备注：: 16 pages, 5 figures

摘要：

给定一个整数$k$，将$C_k$定义为整数集合$n > \max(k,0)$，使得$a^{n-k+1} \equiv a \pmod{n}$对所有整数$a$成立。建立了$C_k$中元素的各种乘法性质，并给出了$C_k$无限的一个充分条件.此外，我们还证明了有一个和两个素因子的$C_k$中有多个元素当且仅当$k>0$和$k$是素的.此外，如果$n \in C_k$的除了两个素因子之外的所有素因子都是固定的，那么除了某些$n$的无限族之外，$C_k$中有很多元素。我们还用数值证据给出了有关$C_k$增长率的猜测。当$a$和$k$都是固定的时，我们探讨了一个类似的问题，并证明了对于固定整数$a \geq 2$和$k$，有无穷多个整数$n$使得$a^{n-k} \equiv 1 \pmod{n}$当且仅当$(k,a) \neq (0,2)$通过构建Kiss和Phong的工作。最后讨论了正整数$n$使得Carmichael函数$λ(n)$整除$n-k$的乘法性质.

英文摘要：

Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$ such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient condition for the infinitude of $C_k$. Moreover, we prove that there are finitely many elements in $C_k$ with one and two prime factors if and only if $k>0$ and $k$ is prime. In addition, if all but two prime factors of $n \in C_k$ are fixed, then there are finitely many elements in $C_k$, excluding certain infinite families of $n$. We also give conjectures about the growth rate of $C_k$ with numerical evidence. We explore a similar question when both $a$ and $k$ are fixed and prove that for fixed integers $a \geq 2$ and $k$, there are infinitely many integers $n$ such that $a^{n-k} \equiv 1 \pmod{n}$ if and only if $(k,a) \neq (0,2)$ by building off the work of Kiss and Phong. Finally, we discuss the multiplicative properties of positive integers $n$ such that Carmichael function $λ(n)$ divides $n-k$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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