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Carmichael Numbers in All Possible Arithmetic Progressions

Daniel Larsen

TL;DR

The paper resolves a long-standing question about the distribution of Carmichael numbers in arithmetic progressions by proving that every Carmichael-compatible progression contains infinitely many Carmichael numbers. It introduces a two-set primes method, extending the Alford–Granville–Pomerance construction, to build Carmichael seeds constrained by multiple modular conditions and then combines these seeds with primes of the form $d k+1$ to satisfy Korselt's criterion. The authors obtain a quantitative lower bound $x^{1/168-\\epsilon}$ for Carmichael numbers in compatible progressions and deduce that $\liminf_{n\text{ Carmichael}} \frac{φ(n)}{n}=0$, answering a question of Alford–Granville–Pomerance. The approach hinges on refined distribution results for primes in APs, a probabilistic construction of the divisor set $L$, two interacting prime families, and large-sieve/character methods to control subgroup-avoidance and residue constraints, culminating in a robust framework for generating Carmichael numbers in a wide class of progressions.

Abstract

We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if $m$ is any integer such that $(m,2φ(m))=1$ then there exist infinitely many Carmichael numbers divisible by $m$. As a consequence, we are able to prove that $\liminf_{n\text{ Carmichael}}\frac{φ(n)}{n}=0$, resolving a question of Alford, Granville, and Pomerance.

Carmichael Numbers in All Possible Arithmetic Progressions

TL;DR

The paper resolves a long-standing question about the distribution of Carmichael numbers in arithmetic progressions by proving that every Carmichael-compatible progression contains infinitely many Carmichael numbers. It introduces a two-set primes method, extending the Alford–Granville–Pomerance construction, to build Carmichael seeds constrained by multiple modular conditions and then combines these seeds with primes of the form to satisfy Korselt's criterion. The authors obtain a quantitative lower bound for Carmichael numbers in compatible progressions and deduce that , answering a question of Alford–Granville–Pomerance. The approach hinges on refined distribution results for primes in APs, a probabilistic construction of the divisor set , two interacting prime families, and large-sieve/character methods to control subgroup-avoidance and residue constraints, culminating in a robust framework for generating Carmichael numbers in a wide class of progressions.

Abstract

We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if is any integer such that then there exist infinitely many Carmichael numbers divisible by . As a consequence, we are able to prove that , resolving a question of Alford, Granville, and Pomerance.

Paper Structure

This paper contains 9 sections, 27 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Primes in Arithmetic Progressions
  4. Constructing $L$
  5. Constructing Families of Primes
  6. Constructing Carmichael Numbers
  7. Escaping Subgroups
  8. Constructing Products with Specified Residues
  9. Constructing Carmichael Numbers from Carmichael Seeds

Key Result

Theorem 1

Let $r\ (\text{mod}\ m)$ be a Carmichael compatible arithmetic progression. Then for every $\epsilon>0$ and for every $x$ sufficiently large in terms of $\epsilon$, there are more than $x^{1/168-\epsilon}$ Carmichael numbers less than $x$ congruent to $r$ mod $m$.

Theorems & Definitions (52)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 42 more