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Primes in LCM recurrences: a density theorem via companion sieves

Benoit Cloitre

Abstract

Let $(a_n)$ be defined by $a_1=1$ and $a_n=a_{n-1}+\lcm(n,a_{n-1})$. The multiplicative increments $b_n = n/\gcd(n,a_{n-1})$ are conjectured to be either $1$ or prime. Unlike Rowland's GCD-based sequence where elementary techniques suffice, pointwise proofs for the LCM variant encounter the Linnik barrier at $L=2$. We introduce the Companion--Sieve framework. It bypasses pointwise arguments by analyzing a guaranteed factor reservoir on average. Using the Bombieri-Vinogradov theorem and elementary sieve bounds, we prove unconditionally that the conjecture holds for a set of integers of asymptotic density $1$. A variant of the LCM recurrence reveals an inhibition mechanism that dynamically encodes the Twin Prime Conjecture, illustrating how LCM-driven recurrences connect to classical problems in multiplicative number theory.

Primes in LCM recurrences: a density theorem via companion sieves

Abstract

Let be defined by and . The multiplicative increments are conjectured to be either or prime. Unlike Rowland's GCD-based sequence where elementary techniques suffice, pointwise proofs for the LCM variant encounter the Linnik barrier at . We introduce the Companion--Sieve framework. It bypasses pointwise arguments by analyzing a guaranteed factor reservoir on average. Using the Bombieri-Vinogradov theorem and elementary sieve bounds, we prove unconditionally that the conjecture holds for a set of integers of asymptotic density . A variant of the LCM recurrence reveals an inhibition mechanism that dynamically encodes the Twin Prime Conjecture, illustrating how LCM-driven recurrences connect to classical problems in multiplicative number theory.
Paper Structure (40 sections, 22 theorems, 61 equations, 1 table)

This paper contains 40 sections, 22 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction: from Rowland's simplicity to LCM's depth
  2. The landscape of prime-generating recurrences
  3. The LCM variant
  4. Why LCM is fundamentally harder
  5. Main results and the companion-sieve philosophy
  6. Methodological stance.
  7. Organization of the paper
  8. Preliminaries and standard inputs
  9. Notation
  10. Basic properties of the sequence
  11. Prime distribution in arithmetic progressions
  12. Elementary sieve bounds
  13. The Bombieri--Vinogradov theorem
  14. Comparison with Ruiz-Cabello's approach
  15. Ruiz-Cabello's pointwise strategy
  16. ...and 25 more sections

Key Result

Theorem 1.2

The set of integers $n$ for which $b_n\in\{1\}\cup\mathbb{P}$ has asymptotic density $1$.

Theorems & Definitions (50)

  • Conjecture 1.1: Main LCM conjecture
  • Theorem 1.2: Unconditional density 1
  • Theorem 1.3: Effective finite reduction
  • Theorem 1.4: Twin Prime encoding
  • Remark 1.5: Companion sieve vs auxiliary sieve
  • Remark 2.1: Standard notation
  • Lemma 2.2: Basic bounds and parity
  • proof
  • Theorem 2.3: Siegel--Walfisz; e.g. Davenport
  • Lemma 2.4: Mertens' estimate; e.g. MV Th. 2.7
  • ...and 40 more