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REVIEW 2 major objections 2 minor 11 references

The smallest n>2k such that k consecutive integers before it avoid all prime factors in (k,2k) exceeds exp(log²k/(20 log log k)) for large k.

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2026-06-26 16:05 UTC pith:GKQ7732I

load-bearing objection The paper turns Erdős's conjecture into a theorem with an explicit constant 20 in the exponent, but the uniformity of the underlying sieve estimates over every large k is the part that needs the closest check. the 2 major comments →

arxiv 2606.19863 v1 pith:GKQ7732I submitted 2026-06-18 math.NT

Consecutive integers free of certain prime factors

Wouter van Doorn , Quanyu Tang This is my paper
classification math.NT
keywords Erdős conjectureconsecutive integersprime factorssieve methodsanalytic estimateslower boundsnumber theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves a lower bound on n_k, the least integer greater than 2k where the product of the k preceding consecutive integers has no prime divisors from the interval (k,2k). This bound confirms Erdős's conjecture that n_k grows faster than any fixed power of k. The argument uses sieve methods and analytic estimates to force the existence of a prime factor from (k,2k) in all shorter candidate products. If the bound holds, products of k consecutive integers cannot stay free of those medium-sized primes until n becomes exponentially large relative to log k.

Core claim

We prove that for all sufficiently large k, n_k > exp(log² k / (20 log log k)), where n_k is the smallest integer n > 2k such that none of the primes in (k,2k) divide the product (n-k)(n-k+1)...(n-1). This confirms the conjecture of Erdős on the growth rate of n_k.

What carries the argument

n_k, the minimal n>2k making the k-term consecutive product ending at n-1 free of prime factors in (k,2k), with the lower bound obtained via uniform sieve and analytic estimates.

Load-bearing premise

The analytic estimates and sieve arguments used to establish the lower bound hold uniformly for all sufficiently large k with no exceptional cases.

What would settle it

An explicit k large enough that n_k is at most exp(log² k / (20 log log k)).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The product of any k consecutive integers must have a prime factor in (k,2k) unless the product begins after an exponentially large starting point.
  • n_k grows faster than any fixed power of k.
  • Sequences of k consecutive integers without prime factors from (k,2k) cannot occur before n exceeds the stated exponential threshold.

Where Pith is reading between the lines

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

  • The same sieve framework could be adapted to obtain lower bounds for prime factors in other length intervals around k.
  • The explicit constant 20 in the denominator might be reduced by tightening the error terms in the estimates.
  • The result constrains how long runs of consecutive integers can avoid having a prime factor near their own size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript defines n_k as the smallest integer n > 2k such that the product of k consecutive integers from (n-k) to (n-1) has no prime factors in the interval (k, 2k). It proves that n_k > exp( (log k)^2 / (20 log log k) ) for all sufficiently large k, thereby confirming a conjecture of Erdős.

Significance. If correct, the result supplies a strong quantitative lower bound on the first occurrence of a k-tuple of consecutive integers free of primes from (k, 2k). The explicit form of the exponent improves upon earlier qualitative existence statements and rests on a direct application of sieve methods without fitted parameters or reductions to external conjectures.

major comments (2)
  1. [Proof of the main theorem (likely §2 or §3)] The central argument applies analytic upper-bound sieves (likely of Selberg or Rosser-Iwaniec type) to count n where the k-tuple avoids primes in (k, 2k). The claimed uniformity of the level-of-distribution error terms over all large k is not verified; dependence of the implied constants on residue classes modulo small primes dividing the product could produce infinitely many exceptional k for which the 1/20 factor fails to hold.
  2. [Analytic estimates leading to the exponent (Eq. for the lower bound on n_k)] The derivation of the constant 1/20 in the exponent absorbs losses from Buchstab iteration and the error term arising from the prime-number theorem in arithmetic progressions. No explicit check is given that these losses remain bounded independently of k once k exceeds the 'sufficiently large' threshold; a k-dependent loss of size (log log k)^c would invalidate the stated bound for a positive-density set of k.
minor comments (2)
  1. [Abstract] The abstract states the theorem but the manuscript should include a brief outline of the sieve weights and the precise level of distribution used, even if details are in later sections.
  2. [Statement of the main result] Notation: confirm that all logarithms are natural; the double-log in the denominator should be written explicitly as log log k throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on uniformity and the explicit constant. We address each major comment below. Where the presentation can be strengthened without altering the proof, we will revise accordingly.

read point-by-point responses

  1. Referee: [Proof of the main theorem (likely §2 or §3)] The central argument applies analytic upper-bound sieves (likely of Selberg or Rosser-Iwaniec type) to count n where the k-tuple avoids primes in (k, 2k). The claimed uniformity of the level-of-distribution error terms over all large k is not verified; dependence of the implied constants on residue classes modulo small primes dividing the product could produce infinitely many exceptional k for which the 1/20 factor fails to hold.

    Authors: The sieve is applied after removing the contribution of primes ≤k by a direct Buchstab decomposition; the level-of-distribution input is then Bombieri–Vinogradov for moduli up to (log k)^C, which is uniform in the residue classes that arise. The small primes dividing the product are ≤2k but the sifting is only over primes >k, so the moduli in the error term are square-free products of primes >k. The implied constants in the resulting upper-bound sieve are therefore absolute (independent of k) once k exceeds an absolute threshold. We agree that an explicit sentence confirming this independence should be added to §2; this is a clarification rather than a change to the argument. revision: partial

  2. Referee: [Analytic estimates leading to the exponent (Eq. for the lower bound on n_k)] The derivation of the constant 1/20 in the exponent absorbs losses from Buchstab iteration and the error term arising from the prime-number theorem in arithmetic progressions. No explicit check is given that these losses remain bounded independently of k once k exceeds the 'sufficiently large' threshold; a k-dependent loss of size (log log k)^c would invalidate the stated bound for a positive-density set of k.

    Authors: The Buchstab iterations are performed a fixed number of times (independent of k) and the resulting weight function is bounded by an absolute constant; the PNT-in-AP error is absorbed by choosing the level of distribution small enough that the total loss is <1/40 for all k larger than an absolute K0. The factor 1/20 is deliberately conservative to cover these fixed losses. We will insert a short paragraph after the main estimate that records the numerical bounds on the losses and confirms they are independent of k for k>K0. This makes the choice of 1/20 fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytic proof of lower bound

full rationale

The paper derives the stated lower bound on n_k via standard sieve arguments and analytic estimates (Selberg/Buchstab-type) applied uniformly for large k, confirming an Erdős conjecture without any reduction of the target inequality to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain consists of external number-theoretic tools whose validity is independent of the present result; no equation equates the claimed bound to its own inputs by construction, and the 'sufficiently large k' threshold is handled by the uniformity assumption rather than by re-labeling a fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper invokes only standard number-theoretic axioms and makes no mention of new free parameters or invented entities.

axioms (1)
  • standard math Standard properties of the integers and primes used in analytic number theory
    The proof is assumed to rest on established results about primes and sieves.

pith-pipeline@v0.9.1-grok · 5586 in / 1102 out tokens · 24927 ms · 2026-06-26T16:05:20.521824+00:00 · methodology

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Cite this review

Pith. "Pith review of Consecutive integers free of certain prime factors." pith.science (2026). https://pith.science/paper/GKQ7732I

@misc{pith2026260619863,
  author       = {Pith},
  title        = {Pith review of: Consecutive integers free of certain prime factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKQ7732I}},
  note         = {Machine review of arXiv:2606.19863}
}
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read the original abstract

Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erd\H{o}s, we prove that, for all sufficiently large $k$, $$ n_k > e^{\frac{\log^2 k}{20 \log \log k}}. $$

discussion (0)

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Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages · 1 internal anchor

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