On Generalized Carmichael Numbers

Abstract

Given an integer $k$, define $C_k$ as the set of integers $n>\max (k,0)$ such that $a^{n-k+1}\equiv a(\mathrm{mod}n)$ 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 $a\ge 2$ and $k$, there are infinitely many integers $n$ such that $a^{n-k}\equiv 1(\mathrm{mod}n)$ if and only if $(k,a)\ne (0,2)$ by building off the work of Kiss and Phong. Finally, we discuss the multiplicative properties of positive integers $n$ such that Carmichael function $\lambda (n)$ divides $n-k$.

16 pages, 5 figures

References in corpus (1)

• Infinitely Many Carmichael Numbers in Arithmetic Progressions