Carmichael Numbers with Prime Numbers of Prime Factors

TL;DR

This paper proves, assuming Heath-Brown's conjecture, that infinitely many Carmichael numbers have a prime number of prime factors, advancing understanding of their distribution and properties.

Contribution

It establishes the infinitude of Carmichael numbers with a prime number of prime factors under a key conjecture, a novel result in number theory.

Findings

Infinitely many Carmichael numbers with prime number of prime factors are proven to exist.

The proof relies on Heath-Brown's conjecture on the first prime in an arithmetic progression.

The result links the structure of Carmichael numbers to prime distribution conjectures.

Abstract

Under the assumption of Heath-Brown's conjecture on the first prime in an arithmetic progression, we prove that there are infinitely many Carmichael numbers $n$ such that the number of prime factors of $n$ is prime.