Arithmetic progressions of Carmichael numbers in a reduced residue class

TL;DR

Under the assumption of the Prime $k$-tuple Conjecture, the paper proves the existence of arbitrarily long arithmetic progressions of Carmichael numbers within a specific residue class, each composed of three distinct primes.

Contribution

It demonstrates, assuming a major conjecture, the existence of long arithmetic progressions of Carmichael numbers in a fixed residue class, with each number being a product of three primes.

Findings

Existence of arbitrarily long arithmetic progressions of Carmichael numbers in a residue class

Carmichael numbers can be composed of three distinct primes in these progressions

Results depend on the unproven Prime $k$-tuple Conjecture

Abstract

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a product of three distinct prime numbers.