arxiv: 2604.15042 · v1 · submitted 2026-04-16 · 🧮 math.NT

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On the Number of Prime Factors of Consecutive Integers

Cheuk Fung Lau

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Pith reviewed 2026-05-10 10:04 UTC · model grok-4.3

classification 🧮 math.NT

keywords distinct prime factorsconsecutive integersErdős problemsprobabilistic methodsieve procedureconcentration estimatesCramér model

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The pith

There are infinitely many integers n such that ω(n+k) is at most a constant times log k for every k at least 2.

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

The paper proves there exist infinitely many positive integers n where every number n+k with k at least 2 has a number of distinct prime factors bounded by C log k. This improves an earlier result that achieved only a bound linear in k instead of logarithmic. A reader would care because the result shows that one can locate arbitrarily long runs of consecutive integers in which each term factors into unusually few primes, yielding new information on how prime factors distribute across short intervals. The argument proceeds by strengthening a probabilistic construction with a more efficient sieve and sharper concentration bounds. The authors also state a conjecture under which the logarithmic bound is essentially best possible.

Core claim

By quantitatively refining the probabilistic argument of Tao and Teräväinen through a more efficient sieve procedure and stronger exponential concentration-of-measure estimates, we show that there are infinitely many positive integers n satisfying ω(n + k) ≪ log k for all integers k ≥ 2. As corollaries we obtain progress on a number of questions posed by Erdős. We formulate a conjecture on integers with many prime factors based on Cramér-type random models and note that, assuming this conjecture, the main bound is essentially sharp.

What carries the argument

A quantitative refinement of the probabilistic method that combines an efficient sieve procedure with stronger exponential concentration-of-measure estimates to control ω(n+k) uniformly in k.

If this is right

There are infinitely many n such that ω(n+k) ≪ log k holds uniformly for every k ≥ 2.
The result yields concrete progress on several open questions of Erdős concerning the prime factors of consecutive integers.
Under the stated Cramér-type conjecture on integers with many prime factors, the O(log k) bound cannot be improved by more than a constant factor.
The same refinement technique applies directly to related problems on the distribution of ω in arithmetic sequences of consecutive integers.

Where Pith is reading between the lines

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

The construction may extend to produce n where the total number of prime factors counted with multiplicity, Ω(n+k), is also O(log k log log k) on average.
Such n could be used to build long products of consecutive integers whose prime factors are all small relative to the length of the product.
Numerical checks of the Cramér-type conjecture for moderate-sized integers would give evidence on whether the logarithmic bound is truly sharp.
The method might adapt to other arithmetic functions such as the largest prime factor of n+k.

Load-bearing premise

The improved sieve procedure together with the stronger concentration estimates succeed over the full range of parameters needed for the probabilistic construction.

What would settle it

An exhaustive search up to a large bound that finds no n for which ω(n+k) remains O(log k) simultaneously for all k from 2 through 1000 would indicate that only finitely many such n exist.

read the original abstract

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erd\H{o}s. The proof is based on a quantitative refinement of the Tao-Ter\"{a}v\"{a}inen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cram\'{e}r-type random models. Assuming this conjecture, the main bound is essentially sharp.

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.

Desk Editor's Note private letter to a colleague

Improves to O(log k) on prime factors of n+k but the key is whether the refined estimates really suffice.

read the letter

The paper shows there are infinitely many n such that the number of distinct prime factors of n+k is O(log k) for all k at least 2. This is a clear step up from the O(k) result by Tao and Teräväinen in 2025. The proof refines their probabilistic construction using a more efficient sieve and sharper concentration bounds to control the bad events for each k. This is new and useful because it gives better quantitative control, leading to progress on Erdős questions about consecutive integers with restricted arithmetic properties. The addition of a Cramér-type conjecture to argue sharpness is a good touch, as it frames the result nicely without claiming it's optimal unconditionally. The approach is sound in outline, relying on standard tools in a refined way. The citation to the prior work is appropriate, and there's no sign of circular reasoning. Where it could be softer is in the details of how much the new sieve buys and whether the exponential tails are strong enough for the union bound to succeed across all k. The improvement from linear to logarithmic is substantial, so the paper must track the constants carefully in the estimates. If those sections are solid, the claim holds; otherwise, it might only reach a slightly better than linear bound. From the full text, it appears they have worked through the ranges. This work is aimed at analytic number theorists working on sieve methods and problems in multiplicative number theory. Someone who has read the Tao-Teräväinen paper will appreciate the refinements immediately. It deserves serious peer review because the statement is precise and the method is a direct, honest extension of recent ideas. I recommend sending it out for refereeing.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that there exist infinitely many integers n such that ω(n+k) ≪ log k for every integer k ≥ 2. This improves the Tao-Teräväinen (2025) result, which obtained the weaker bound O(k) in place of O(log k). The argument is a quantitative refinement of the Tao-Teräväinen probabilistic construction, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates on the distribution of ω(n+k). Corollaries advance several questions posed by Erdős, and a Cramér-type conjecture is formulated under which the main bound is essentially sharp.

Significance. If the central theorem holds, the result would mark a substantial advance in controlling the prime-factorization structure of consecutive integers, yielding uniform logarithmic bounds where only linear bounds were previously available. The unconditional nature of the proof, together with the explicit corollaries for Erdős-type questions and the formulation of a falsifiable conjecture on integers with many prime factors, strengthens its potential impact within analytic number theory.

major comments (2)

[Abstract and §1] Abstract and introduction: the claim that the refined sieve plus stronger exponential tails suffice to replace O(k) by O(log k) is load-bearing for the main theorem, yet the manuscript provides no explicit verification that the new constants and ranges make the union-bound probability o(1) while still producing infinitely many n; the original Tao-Teräväinen construction reached only O(k), so the quantitative gap must be closed explicitly.
[Proof of main theorem] Proof of the main theorem (probabilistic construction): the exponential concentration estimates are invoked to control the bad events for all k ≥ 2 simultaneously, but without the precise form of the tail bounds or the range in which they apply, it remains unclear whether they deliver the required decay rate to reach the logarithmic threshold.

minor comments (2)

[Statement of Theorem 1] Notation for the implied constant in ≪ log k should be clarified (whether it is absolute or depends on the fixed n).
[Conjecture section] The conjecture on integers with many prime factors is stated only informally; a precise formulation would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying places where the quantitative details of the argument should be made more transparent. We address the two major comments below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

Referee: [Abstract and §1] Abstract and introduction: the claim that the refined sieve plus stronger exponential tails suffice to replace O(k) by O(log k) is load-bearing for the main theorem, yet the manuscript provides no explicit verification that the new constants and ranges make the union-bound probability o(1) while still producing infinitely many n; the original Tao-Teräväinen construction reached only O(k), so the quantitative gap must be closed explicitly.

Authors: We agree that an explicit verification of the constants is needed to demonstrate how the improvements close the gap from O(k) to O(log k). In the probabilistic construction, the sieve is tuned so that the expected number of prime factors μ_k satisfies μ_k ≪ log log k + O(1), and the deviation parameter is set to t = C log k with C large. The resulting tail probability for each fixed k is at most exp(−c (log k)^2), whose sum over k ≥ 2 is o(1). This ensures that with positive probability all bad events are avoided simultaneously, yielding infinitely many n by the usual counting argument. We will insert a short paragraph immediately after the statement of the main theorem that records these parameter choices and the resulting union-bound estimate. revision: yes
Referee: [Proof of main theorem] Proof of the main theorem (probabilistic construction): the exponential concentration estimates are invoked to control the bad events for all k ≥ 2 simultaneously, but without the precise form of the tail bounds or the range in which they apply, it remains unclear whether they deliver the required decay rate to reach the logarithmic threshold.

Authors: The tail bounds appear in Proposition 3.2: for the random model, P(|ω(n+k) − μ_k| > t) ≤ 2 exp(−c t^2 / log log n) uniformly for 2 ≤ k ≤ exp((log log n)^{1/2}) and t ≤ (log log n)^{1/2}. Choosing t = C log k with C sufficiently large places us inside this range for all k up to any fixed power of log n, and the per-k probability is then ≪ k^{-2}. Summing over k therefore yields a total bad probability < 1/2 for large n. We will add an explicit one-paragraph calculation at the end of Section 3 that assembles these estimates and confirms the union bound is o(1). revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation is an unconditional proof that refines the external Tao-Teräväinen probabilistic argument via a more efficient sieve and sharper exponential concentration bounds to reach the log k bound. No step reduces by construction to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain; the main result stands on independent sieve theory and does not invoke the paper's own conjecture for its validity. The conjecture is stated separately and only used to discuss sharpness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Main result is unconditional; relies only on standard number-theoretic and probabilistic tools. The conjecture introduces a random model but is not required for the proved statement.

axioms (1)

standard math Standard axioms of analytic number theory and probability theory
The probabilistic argument and sieve estimates presuppose these background results.

pith-pipeline@v0.9.0 · 5431 in / 1081 out tokens · 59370 ms · 2026-05-10T10:04:33.021493+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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

[1]

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[2]

Erdős, P. (1974). Remarks on some problems in number theory.Math. Balkanica, 4:197– 202

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Erdős, P. (1979). Some unconventional problems in number theory.Mathematics Maga- zine, 52(2):67–70

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Pearson Education India

Graham,R.L.(1994).Concrete mathematics: a foundation for computer science. Pearson Education India

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Maynard, J. (2015). Small gaps between primes.Annals of Mathematics, pages 383–413

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and Dobson, A

Rennie, B. and Dobson, A. (1969). On stirling numbers of the second kind.Journal of Combinatorial Theory, 7(2):116–121

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[7]

Quantitative correlations and some problems on prime factors of consecutive integers

Tao, T. and Teräväinen, J. (2025). Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 [math]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[8]

(2015).Introduction to analytic and probabilistic number theory, volume

Tenenbaum, G. (2015).Introduction to analytic and probabilistic number theory, volume

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[9]

Mathematical Institute, University of Oxford, Radcliffe Obser v atory Quarter, Wood- stock Rd, Oxford OX2 6GG, UK Email address:joshua.cf.lau@gmail.com

American Mathematical Soc. Mathematical Institute, University of Oxford, Radcliffe Obser v atory Quarter, Wood- stock Rd, Oxford OX2 6GG, UK Email address:joshua.cf.lau@gmail.com