形如 $2^n p^m+1$ 且指数 $m$ 为奇数的 Carmichael 数：Florian Luca 定理的改进、一个一般独立性引理及计算研究

中文摘要 AI 辅助

Florian Luca 将他一生中的最后一篇论文献给了集合 $\mathcal K$，该集合由所有奇数正整数 $k$ 组成，使得对于某个正整数 $n$，$2^n k+1$ 是 Carmichael 数。他证明了若 $m\ge 5$ 固定且为奇数，则只有有限多个素数 $p$ 满足 $p^m\in\mathcal K$。在本文中，作为对他记忆的小小致敬，我们仅以他已发表的工作为参考基础，从三个方向推广了他的结果。首先，我们从他的论证中提取出显式的定量界：每个 Carmichael 数 $N=2^n p^m+1$（其中 $m$ 为奇数）满足 $n<2^{2\cdot10^7(m+1)^2m^2(\log p)^2}$，且若 $p>m$，则更强的界 $n<25m^4\log p$ 成立；对于 $p>500m^5\log m$，我们进一步记录素因子个数的一致上界 $\omega(N)\le 26m^7$。作为应用，不超过 $x$ 的这种 Carmichael 数的个数为 $O_m(x^{1/m})$。其次，我们将 Luca 证明中的乘法独立性步骤从底数 $2$ 推广到任意偶数底数 $b$：若 $b$ 为偶数，$m\ge3$ 为奇数且与 $b$ 互素，$p>m$ 为素数，且 $N=b^n p^m+1$ 是 Carmichael 数，则只要 $b^a p^\beta+1$ 是 $N$ 的素因子，$b^n p^m$ 与 $b^a p^\beta$ 就是乘法独立的。第三，我们报告一个穷举计算，表明唯一满足 $2^n p^m+1\le 10^{18}$ 且奇数 $3\le m\le 37$ 的 Carmichael 数是 $1729=2^6\cdot 3^3+1$，不存在奇数 $5\le m\le 37$ 的这种数（对于 $5\le m\le 15$ 扩展到 $10^{21}$），并且唯一满足偶数 $4\le m\le 36$ 的例子是 $46657=2^6\cdot 3^6+1=13\cdot 37\cdot 97$，其中 $3^6=27^2$ 是 $\mathcal K$ 中最小元素的平方。

英文摘要

Florian Luca devoted the last paper of his life to the set $\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\in\mathcal K$. In this note, written as a small tribute to his memory, we extend his result in three directions, using only his published work as our reference base. First, we extract from his arguments explicit quantitative bounds: every Carmichael number $N=2^n p^m+1$ with $m$ odd satisfies $n<2^{2\cdot10^7(m+1)^2m^2(\log p)^2}$, and if $p>m$ the much sharper bound $n<25m^4\log p$ holds; for $p>500m^5\log m$ we further record the uniform bounds $ω(N)\le 26m^7$ on the number of prime factors. As an application, the number of such Carmichael numbers up to $x$ is $O_m(x^{1/m})$. Second, we generalize the multiplicative independence step of Luca's proof from base $2$ to an arbitrary even base $b$: if $b$ is even, $m\ge3$ is odd and coprime to $b$, $p>m$ is prime and $N=b^n p^m+1$ is Carmichael, then $b^n p^m$ and $b^a p^β$ are multiplicatively independent whenever $b^a p^β+1$ is a prime factor of $N$. Third, we report an exhaustive computation showing that the only Carmichael number $2^n p^m+1\le 10^{18}$ with odd $3\le m\le 37$ is $1729=2^6\cdot 3^3+1$, that no such number exists with odd $5\le m\le 37$ (extended to $10^{21}$ for $5\le m\le 15$), and that the only example with even $4\le m\le 36$ is $46657=2^6\cdot 3^6+1=13\cdot 37\cdot 97$, where $3^6=27^2$ is the square of the smallest element of $\mathcal K$.

发表机构

• University of Kara（卡拉大学）

机构由 AI 辅助整理，请以论文原文为准。