REVIEW 2 major objections 3 minor
Rough numbers between consecutive primes
T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read The paper proves that almost every gap between consecutive primes contains an integer whose least prime factor is at least the gap length, with the count of exceptions at most $O(X/\log^2 X)$.
desk verdict A likely real theorem: first confirmation of Erdős's rough-number-in-prime-gap prediction, with a near-optimal exception bound—but the proof is only visible through the abstract, and the Montgomery–Soundararajan input needs a careful referee. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central condition is the rough-number requirement on the least prime factor $p(m)$ relative to the gap length. The argument translates the absence of such an $m$ into a sieve-counting statement over admissible prime tuples: a gap has no qualifying rough number exactly when certain tuples fail to be prime in a short interval. The singular series of these tuples then enters through an averaging identity that converts the exceptional-gap count into the stated $O(X/\log^2 X)$ bound, and the prime tuples conjecture would sharpen the average to an asymptotic.
What would settle it
Compute $N(X)$ by factoring all integers inside $(p_n,p_{n+1})$ for $p_n\in[X,2X]$ at large $X$ (say $10^{12}$). If $N(X)$ grows like $X/\log X$ rather than $O(X/\log^2 X)$, the unconditional bound is false.
Extended reading notes
Core claim
The prediction is that in almost all prime gaps $(p_n,p_{n+1})$, there is a natural number $m$ with least prime factor $p(m) \ge p_{n+1}-p_n$. The paper establishes this prediction. Let $N(X)$ count exceptional $n$ with $p_n \in [X,2X]$, meaning no such $m$ lies in the gap. The main unconditional result is $N(X) = O(X/\log^2 X)$. Conditionally on a form of the prime tuples conjecture, the paper obtains $N(X) \sim cX/\log^2 X$ for an explicit constant $c>0$ numerically suggested to be about $2.7$--$2.8$. The proof is a sieve-theoretic argument; the full-strength version depends on a previously developed asymptotic for singular series.
Load-bearing premise
The full-strength result relies on a known formula for how often a pattern of primes occurs, and that formula must remain accurate for all the patterns the sieve averages over; if it fails in that range, the $O(X/\log^2 X)$ bound could be too strong.
Editorial extensions
If this is right
- Almost all prime gaps contain a rough number: the proportion of exceptional gaps among $p_n \in [X,2X]$ decays as $O(1/\log X)$, so exceptions are sparse.
- The $O(X/\log^2 X)$ exception bound is within a logarithm of the total number of gaps, which is about $X/\log X$.
- Under the prime tuples conjecture, the exact constant is explicitly given and numerically sits between $2.7$ and $2.8$, so the conjecture can be tested by computing $N(X)$ at large $X$.
- The result confirms the classical prediction in a quantitative form.
Reading between the lines
- A testable extension: compute $N(X)$ numerically up to large $X$ (say $10^{12}$) to compare the empirical constant with the predicted $c\in(2.7,2.8)$; a clear drift outside that interval would point to a problem in the conditional asymptotic's singular-series averaging, not necessarily in the unconditional sieve bound.
- The same rough-number question can be posed for gaps between primes in arithmetic progressions or between other sparse sequences; if the sieve argument is robust, a similar $O(X/\log^2 X)$ bound should hold there.
- The result suggests a sharper structural picture of prime gaps: not only are their lengths often near the average, but they are 'rough' in this specific factor sense; this may interact with heuristics for maximal prime gaps, where the longest gaps are exactly those for which all interior integers have a small prime factor.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims an unconditional sieve-theoretic proof that almost every prime gap (p_n, p_{n+1}) with p_n in [X,2X] contains an integer m whose least prime factor is at least the gap length, with the number of exceptional gaps N(X) = O(X/log^2 X). Under a form of the Hardy--Littlewood prime tuples conjecture, the authors claim the more precise asymptotic N(X) ~ c X/log^2 X for an explicit constant c believed to lie between 2.7 and 2.8. The abstract states that the full-strength results rely on the singular-series asymptotics of Montgomery and Soundararajan. Only the abstract was available for review; the sieve argument, the precise hypotheses, and the derivations could not be inspected.
Significance. If correct, the unconditional bound would be the first proof of Erdős's prediction and would give a near-optimal exceptional-set bound, while the conditional asymptotic with an explicit constant would be a valuable refinement. The reliance on an external analytic-number-theory input (Montgomery--Soundararajan singular-series asymptotics) is a concrete point of potential fragility, but the manuscript's novelty and importance are real if the proof is sound. Because the full text is absent, the significance can only be assessed conditional on verification of the proof details.
major comments (2)
- [Abstract, last sentence] The sentence 'To obtain our results in their full strength we rely on the asymptotics for singular series developed by Montgomery and Soundararajan' identifies the load-bearing external input, but the abstract does not specify the exact singular-series average being used, the range of gap lengths h entering the average, or whether the averaged series is the standard prime-tuples singular series or a nonstandard variant that also encodes the roughness condition on interior integers. If the Montgomery--Soundararajan theorem requires h to be large (e.g., h >= X^epsilon) while the sieve needs all h up to (log X)^C, or if the averaged object is outside the theorem's scope, the O(X/log^2 X) bound and the constant c are unsupported. The manuscript needs to state the precise theorem used and verify its hypotheses directly.
- [Abstract, central claim] It is unclear whether the unconditional result N(X) = O(X/log^2 X) is independent of the Montgomery--Soundararajan asymptotics or whether it inherits a condition from them. The phrase 'full strength' may indicate a weaker unconditional result, but the abstract does not say. Since the paper's headline claim is the unconditional 'almost all gaps' statement, the authors must explicitly delineate which of the stated results are unconditional and which require the Montgomery--Soundararajan or Hardy--Littlewood assumptions. As written, this ambiguity prevents the reader from knowing the logical status of the main theorem.
minor comments (3)
- [Abstract, notation] The notation p(m) for the least prime factor of m conflicts visually with the notation p_n for primes. Consider using P^-(m) or lpf(m) to avoid ambiguity.
- [Abstract, wording] 'which we believe to be between 2.7 and 2.8' is informal for a mathematical constant. The statement should either give a rigorous interval or a numerical approximation with a definite article, e.g., 'we compute c ≈ 2.75' if that is the case.
- [Abstract, asymptotic notation] The phrase 'at most O(X/log^2 X)' is nonstandard; O-notation already includes an absolute constant. Expressing it as 'N(X) = O(X/log^2 X)' or 'N(X) ≪ X/log^2 X' would be cleaner.
Circularity Check
No circularity found in abstract; derivation is self-contained against external sieve benchmarks.
full rationale
This abstract-only review finds no circular step. The unconditional bound N(X) = O(X/log^2 X) is stated to follow from a sieve-theoretic argument, with no fitted parameter and no reliance on the target conclusion. The conditional asymptotic N(X) ~ c X/log^2 X is explicitly derived from the Hardy--Littlewood prime tuples conjecture, which is an independent external conjecture rather than the paper's own claim. The constant c is said to be explicit and derived from singular-series asymptotics due to Montgomery and Soundararajan; that is an external, independently developed input, not a self-citation or a renamed version of the paper's own output. The skeptical concern that the Montgomery--Soundararajan asymptotics may not apply in the needed range is a correctness or applicability risk, not circularity: it concerns whether an external theorem's hypotheses are met, not whether the conclusion is assumed in the premises. No self-citation, no ansatz smuggled in by citation, and no definitional equivalence between the prediction and the inputs appear in the abstract. Thus the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (2)
- domain assumption Hardy-Littlewood prime tuples conjecture
- domain assumption Montgomery-Soundararajan singular series asymptotics
Cite this review
Pith. "Pith review of Rough numbers between consecutive primes." pith.science (2026). https://pith.science/paper/SAKEMWYV
@misc{pith2026250806463,
author = {Pith},
title = {Pith review of: Rough numbers between consecutive primes},
year = {2026},
howpublished = {\url{https://pith.science/paper/SAKEMWYV}},
note = {Machine review of arXiv:2508.06463}
}
abstract
Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap, confirming a prediction of Erd\H{o}s. In fact the number $N(X)$ of exceptional gaps with $p_n \in [X,2X]$ is shown to be at most $O(X/\log^2 X)$. Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic $N(X) \sim c X / \log^2 X$ for an explicit constant $c>0$, which we believe to be between $2.7$ and $2.8$. To obtain our results in their full strength we rely on the asymptotics for singular series developed by Montgomery and Soundararajan.
Reviewed August 5, 2026 · model on record in the stance chip above.
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