Infinitely Many Carmichael Numbers in Arithmetic Progressions

TL;DR

This paper proves that for any coprime integers a and M, there are infinitely many Carmichael numbers congruent to a modulo M, extending understanding of their distribution in arithmetic progressions.

Contribution

It establishes the infinite occurrence of Carmichael numbers in any given arithmetic progression with coprime parameters, a significant advancement in number theory.

Findings

Infinitely many Carmichael numbers exist in any arithmetic progression coprime to the modulus.

The distribution of Carmichael numbers is dense across all coprime residue classes.

The result generalizes previous knowledge about Carmichael numbers and their distribution.

Abstract

In this paper, we prove that for any $a,M\in \mathbb N$ with $(a,M)=1$, there are infinitely many Carmichael numbers $m$ such that $m\equiv a$ mod $M$