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arxiv: 2510.18891 · v3 · submitted 2025-10-18 · 🧮 math.NT

Primes in LCM recurrences

Benoit Cloitre This is my paper

Pith reviewed 2026-05-18 06:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords LCM recurrenceprime-generating sequenceCompanion-Sieve frameworkBombieri-Vinogradov theoremequidistribution in arithmetic progressionstwin primesanalytic number theoryLinnik's theorem
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The pith

A Companion-Sieve framework reduces the LCM recurrence prime-increment conjecture to an equidistribution problem, which Bombieri-Vinogradov resolves for a density-1 set.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines an LCM-based sequence analogous to Rowland's GCD recurrence, where the multiplicative increments are conjectured to be always 1 or prime. It develops a Companion-Sieve framework that converts this conjecture into an equidistribution statement for primes in the class -1 mod p. Applying the Bombieri-Vinogradov theorem then shows the conjecture holds unconditionally for a set of integers of asymptotic density 1. The work also supplies an effective finite reduction that pushes any counterexample past a computable threshold to involve only large prime factors, and treats a related recurrence whose increments encode twin-prime pairs.

Core claim

The central claim is that the Companion-Sieve framework reduces the prime-increment conjecture for the LCM recurrence to an equidistribution statement on primes in the progression -1 mod p; the Bombieri-Vinogradov theorem then implies that the conjecture is true for a set of positive integers with asymptotic density 1. Any counterexample beyond a computable bound must involve only large prime factors. A closely related recurrence encodes twin-prime pairs in its increment pattern, and a conditional density-1 result holds for it under a prime-index detection hypothesis via an upper-bound Selberg sieve estimate.

What carries the argument

The Companion-Sieve framework, which reduces the prime-increment conjecture to an equidistribution problem for primes congruent to -1 modulo p.

If this is right

  • The prime-increment conjecture holds for a set of integers of asymptotic density 1.
  • Any counterexample must lie beyond a computable bound and involve only large prime factors.
  • A related recurrence encodes twin-prime pairs through its increment pattern.
  • Under a prime-index detection hypothesis, the twin-prime version also satisfies a density-1 result via Selberg sieve bounds on twin primes in arithmetic progressions.
  • The analysis produces three new conjectures on the distribution of primes in arithmetic progressions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Stronger equidistribution hypotheses such as Elliott-Halberstam could lift the density-1 result to the full conjecture.
  • Direct computation of the recurrence for many small starting values could confirm the finite-reduction claim in practice.
  • The same reduction technique may apply to other GCD- or LCM-based prime generators to obtain density statements.
  • Links to sieve methods suggest that zero-density estimates could improve the range of the unconditional result.

Load-bearing premise

The Companion-Sieve framework correctly reduces the prime-increment conjecture to an equidistribution statement for primes in the progression -1 mod p.

What would settle it

Discovery of a single starting integer beyond the computable threshold whose increment is a composite number other than 1, or a disproof that primes are equidistributed in the class -1 mod p for some p.

read the original abstract

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a strengthening of Linnik's theorem on the least prime in an arithmetic progression that lies beyond current reach. We develop a Companion--Sieve framework that reduces the conjecture to an equidistribution problem for primes in the progression $-1\bmod p$, and applying the Bombieri--Vinogradov theorem we prove unconditionally that the conjecture holds for a set of integers of asymptotic density $1$. We also give an effective finite reduction showing that any counterexample beyond a computable threshold involves only large prime factors. A closely related recurrence turns out to encode twin prime pairs through its increment pattern, and we prove a conditional density-$1$ result for it under a prime-index detection hypothesis, using an upper-bound Selberg sieve estimate for twin primes in arithmetic progressions. The analysis also leads to three new conjectures on the distribution of primes in arithmetic progressions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies an LCM-based analogue of Rowland's GCD recurrence for generating primes. It conjectures that the multiplicative increments are always 1 or prime, develops a Companion-Sieve framework reducing the conjecture to an equidistribution statement for primes in the progression -1 mod p, and applies the Bombieri-Vinogradov theorem to prove unconditionally that the conjecture holds on a set of asymptotic density 1. It also gives an effective finite reduction for counterexamples, a conditional density-1 result for a related twin-prime encoding recurrence under a prime-index hypothesis using an upper-bound Selberg sieve, and states three new conjectures on primes in arithmetic progressions.

Significance. If the Companion-Sieve reduction holds, the unconditional density-1 theorem is a solid application of Bombieri-Vinogradov to a concrete recurrence problem, giving a large-scale verification of the prime-increment conjecture without needing a full strengthening of Linnik's theorem. The effective reduction to large prime factors and the conditional twin-prime result are additional strengths; the paper explicitly credits the use of established theorems (Bombieri-Vinogradov and Selberg sieve) rather than ad-hoc constructions.

major comments (2)
  1. [section introducing the Companion-Sieve framework and the reduction step] Companion-Sieve framework and reduction step: the argument that every composite multiplicative increment must produce a detectable failure of equidistribution for primes ≡ -1 mod p is load-bearing for the density-1 claim. It is not immediately clear from the reduction whether composite increments arising from prime factors whose distribution is not fully controlled by this single residue class (or whose moduli depend on the growing LCM in a manner exceeding the Bombieri-Vinogradov level of distribution) are excluded; an explicit case analysis or lemma showing completeness of the reduction is needed.
  2. [section applying Bombieri-Vinogradov to the reduced equidistribution problem] Application of Bombieri-Vinogradov: the equidistribution statement to which the conjecture is reduced must be shown to lie within the range where Bombieri-Vinogradov supplies a positive proportion of primes in the relevant arithmetic progressions up to x; the manuscript should state the precise level of distribution required and verify that the error terms from the recurrence do not exceed it.
minor comments (2)
  1. [introduction or recurrence definition] Clarify the precise definition of the multiplicative increment in the LCM recurrence (early section) to avoid ambiguity with the original Rowland sequence.
  2. [final section] The three new conjectures on primes in arithmetic progressions are stated at the end; a brief comparison with existing conjectures (e.g., Hardy-Littlewood or Elliott-Halberstam) would help readers assess their novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to strengthen the presentation of the Companion-Sieve reduction and the application of Bombieri-Vinogradov.

read point-by-point responses

  1. Referee: Companion-Sieve framework and reduction step: the argument that every composite multiplicative increment must produce a detectable failure of equidistribution for primes ≡ -1 mod p is load-bearing for the density-1 claim. It is not immediately clear from the reduction whether composite increments arising from prime factors whose distribution is not fully controlled by this single residue class (or whose moduli depend on the growing LCM in a manner exceeding the Bombieri-Vinogradov level of distribution) are excluded; an explicit case analysis or lemma showing completeness of the reduction is needed.

    Authors: We agree that the completeness of the reduction requires an explicit verification to ensure all composite cases are covered without exceeding the level of distribution. In the revised manuscript we will add a new lemma (Lemma 3.5) that performs a case analysis on composite multiplicative increments. The lemma shows that any such composite increment necessarily introduces a prime factor q for which the sequence fails equidistribution in the class -1 mod q, and that all relevant moduli arising from the growing LCM remain bounded by the current term size, hence lie strictly inside the Bombieri-Vinogradov range of distribution. revision: yes

  2. Referee: Application of Bombieri-Vinogradov: the equidistribution statement to which the conjecture is reduced must be shown to lie within the range where Bombieri-Vinogradov supplies a positive proportion of primes in the relevant arithmetic progressions up to x; the manuscript should state the precise level of distribution required and verify that the error terms from the recurrence do not exceed it.

    Authors: We will revise the relevant section to state explicitly that the required level of distribution is 1/2 - ε for any fixed ε > 0, which is furnished by the Bombieri-Vinogradov theorem. We will also include a short verification that the error terms contributed by the recurrence (arising from the sieve weights and the LCM growth) are O(x (log x)^{-C}) for sufficiently large C and therefore remain smaller than the admissible error term in Bombieri-Vinogradov, guaranteeing a positive proportion of primes in the progression -1 mod p. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for original recurrence; central density-1 result rests on external Bombieri-Vinogradov theorem

full rationale

The paper defines the LCM recurrence via a 2008 self-citation and develops a Companion-Sieve reduction to an equidistribution statement, then invokes the external Bombieri-Vinogradov theorem to obtain the unconditional density-1 result. No step equates a prediction to a fitted input by construction, renames a known result, or makes the main theorem depend on an unverified self-citation chain. The derivation chain is self-contained against external benchmarks (BV, Selberg sieve) and the recurrence definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper draws on standard analytic number theory results rather than introducing new free parameters or postulated entities. The main technical addition is the derived Companion-Sieve reduction.

axioms (2)
  • standard math Bombieri-Vinogradov theorem on the distribution of primes in arithmetic progressions
    Invoked to establish equidistribution for the density-1 result.
  • standard math Upper-bound Selberg sieve estimates for twin primes in arithmetic progressions
    Used for the conditional density-1 result on the twin-prime encoding recurrence.

pith-pipeline@v0.9.0 · 5707 in / 1409 out tokens · 44373 ms · 2026-05-18T06:03:38.937877+00:00 · methodology

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