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Carmichael Numbers with a Specified Number of Prime Factors

Daniel Larsen, Thomas Wright

TL;DR

This work proves that for all sufficiently large $R$ there exists a Carmichael number with exactly $R$ prime factors, extending the AGP framework with an unconditional construction. By developing equidistribution in carefully crafted prime sets $\mathcal{P}$ built from moduli $L$ and $k$, and leveraging Davenport-type bounds and combinatorial lemmas, the authors show large classes of products yield Carmichael numbers with prescribed $\omega(n)$. They handle both even and odd $\omega(n)$ counts, notably fixing a small prime factor to realize odd counts, and then use a covering argument across scales to conclude that every sufficiently large $R$ occurs. The results provide an unconditional route toward realizing exact prime-factor counts in Carmichael numbers, contributing a significant strengthening of the landscape around Korselt-type constructions and equidistribution methods.

Abstract

For every sufficiently large integer $R$, there exists a Carmichael number with exactly $R$ prime factors.

Carmichael Numbers with a Specified Number of Prime Factors

TL;DR

This work proves that for all sufficiently large there exists a Carmichael number with exactly prime factors, extending the AGP framework with an unconditional construction. By developing equidistribution in carefully crafted prime sets built from moduli and , and leveraging Davenport-type bounds and combinatorial lemmas, the authors show large classes of products yield Carmichael numbers with prescribed . They handle both even and odd counts, notably fixing a small prime factor to realize odd counts, and then use a covering argument across scales to conclude that every sufficiently large occurs. The results provide an unconditional route toward realizing exact prime-factor counts in Carmichael numbers, contributing a significant strengthening of the landscape around Korselt-type constructions and equidistribution methods.

Abstract

For every sufficiently large integer , there exists a Carmichael number with exactly prime factors.
Paper Structure (6 sections, 9 theorems, 29 equations)

This paper contains 6 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Establishing Equidistribution
  4. Finding Primes
  5. Producing Carmichael Numbers
  6. Constructing Carmichael Numbers With Odd Numbers of Prime Factors

Key Result

Theorem 1

For all sufficiently large $R$, there exist Carmichael numbers with precisely $R$ prime factors.

Theorems & Definitions (15)

  • Theorem 1
  • Proposition 1
  • Definition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • Lemma 1
  • ...and 5 more