发表机构
Wofford College（沃福德学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究(a,a)-卡迈克尔数，通过定义K及C_ν(X,a)，在关于算术级数中首个素数的强猜想下，证明对特定整数a和自然数ν，给出C_ν(X,a)的下界。
AI 中文摘要
定义一个(a,a)-卡迈克尔数为无平方因子自然数n，使得p|n蕴含p - a|n - a。对于有素因子p_1,\cdots,p_m的这样一个数n，定义K = GCD[p_1 - a,\cdots,p_m - a]，并设C_ν(X,a)表示不超过X的使得K = ν的(a,a)-卡迈克尔数的个数。假设关于算术级数中首个素数的一个强猜想，我们证明对于任意整数a以及任意满足(ν,a)=1且a和ν奇偶性相反的自然数ν，有C_ν(X,a)≥X^{1-(2 + o(1))\frac{\log\log\log\log X}{\log\log\log X}}。这与许多传统卡迈克尔数构造不同，传统构造通常要求K随n增长。
英文摘要
Define an $(a,a)$-Carmichael number to be a squarefree natural number $n$ such that $p\mid n$ implies $p-a\mid n-a$. For such a number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-a,\cdots,p_m-a],$$ and let $C_ν(X,a)$ denote the number of $(a,a)$-Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any integer $a$ and for any natural number $ν$ with $(ν,a)=1$ and $a$ and $ν$ having opposite parity, $$C_ν(X,a)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from many traditional constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.
CommentsThis is a significant revision and generalization of arXiv:2409.16397, since that paper is now largely subsumed by Larsen's result on Carmichael numbers in arithmetic progressions
