$(a,a)$-Carmichael numbers and greatest common divisors of $p-a$

Abstract

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 $C_\nu(X,a)$ denote the number of $(a,a)$-Carmichael numbers up to $X$ such that $K=\nu$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any integer $a$ and for any natural number $\nu$ with $(\nu,a)=1$ and $a$ and $\nu$ having opposite parity, $$C_\nu(X,a)\geq X^{1-(2+o(1))\frac{\log\log\log \log X}{\log\log\log X}}.$$ This is a departure from many traditional constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.