Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Carmichael Numbers with Prime Numbers of Prime Factors

Thomas Wright

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.

Carmichael Numbers with Prime Numbers of Prime Factors

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 such that the number of prime factors of is prime.
Paper Structure (10 sections, 5 theorems, 34 equations)

This paper contains 10 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Conditional Results
  3. Introduction: Outline
  4. Constructing an $L$
  5. Bounds for primes in arithmetic progressions
  6. Small Orders mod $L$
  7. The Extra Prime
  8. Sizes of Sets
  9. Remarks
  10. Acknowledgements

Key Result

Theorem 2.1

Let $G$ be an abelian group, and let $r>t>n=n(G)$ be integers. Then any sequence of $r$ elements in $G$ must contain $\left(\right)/\left(\right)$ distinct subsequences of length at most $t$ and at least $t-n$ whose product is the identity.

Theorems & Definitions (11)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Theorem 2.1: Alford, Granville, Pomerance 1994
  • Lemma 3.1
  • Lemma 6.1
  • Theorem 7.1
  • proof
  • Theorem 7.2
  • proof
  • ...and 1 more