arXiv 1902.112832019-02math.NT

关于本原卡迈克尔数

On primary Carmichael numbers

Bernd C. Kellner

备注：: 32 pages, 12 tables, final revised version
发表信息：: Integers 22 (2022), Article A38, 1-39
DOI：: 10.5281/zenodo.10963985

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AI总结：

本研究围绕本原卡迈克尔数展开，证明三因子卡迈克尔数构造多项式$U_3(t)$的特殊分解性质，明确其成为本原卡迈克尔数的条件，还探讨了与的士数、多边形数的关联。

摘要：

本原卡迈克尔数是近期被提出的卡迈克尔数的一个特殊子集。一个本原卡迈克尔数$m$具有唯一性质：对其每个素因子$p$，都有$s_p(m)=p$成立，其中$s_p(m)$是$m$在$p$进制下的各位数字之和。第一个这样的数是拉马努金著名的的士数$1729$。根据Chernick的研究，所有三因子卡迈克尔数都可以由某些无平方因子多项式$U_3(t) \in \mathbb{Z}[t]$构造，其中最简单的是$U_3(t)=(6t+1)(12t+1)(18t+1)$。我们证明，对所有$t \geq 2$，任意$U_3(t)$的值都满足一种特殊分解，除了某些例外情况，$t=1$时也满足该分解。这些情况进一步表明，若$U_3(t)$的三个因子同时为奇素数，则$U_3(t)$不仅是卡迈克尔数，还是本原卡迈克尔数。结合例外情况，所有三因子卡迈克尔数至少满足：对其最大素因子$p$，有$s_p(m)=p$成立。随后，我们展示了其与的士数和多边形数的一些联系，再次以$1729$为例。

英文摘要：

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number $m$ has the unique property that $s_p(m) = p$ holds for each prime factor $p$, where $s_p(m)$ is the sum of the base-$p$ digits of $m$. The first such number is Ramanujan's famous taxicab number $1729$. Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials $U_3(t) \in \mathbb{Z}[t]$, the simplest one being $U_3(t) = (6t+1)(12t+1)(18t+1)$. We show that the values of any $U_3(t)$ obey a special decomposition for all $t \geq 2$ and besides certain exceptions also in the case $t=1$. These cases further imply that if all three factors of $U_3(t)$ are simultaneously odd primes, then $U_3(t)$ is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that $s_p(m) = p$ holds for the greatest prime factor $p$ of $m$. Subsequently, we show some connections to taxicab and polygonal numbers, involving the number $1729$ as an example again.

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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摘要：

英文摘要：

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