Long runs of integers with small prime factors and the divisor function of $n!$

Tristan Freiberg

REVIEW

Long runs of integers with small prime factors and the divisor function of $n!$

Tristan Freiberg

Number Theory · Submitted 14 Sep 2026 · 20 references

Pith's verdict: Solid improvement of a bound, with a neat averaging trick; the proof is rigorous and the exposition is clear.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 15, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2609.15597 (version 1)

Pith IDKUWAXOIY, Pith's permanent record of this paper and its review

CitationTristan Freiberg. "Long runs of integers with small prime factors and the divisor function of $n!$." arXiv:2609.15597 (2026). https://pith.science/paper/KUWAXOIY

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This paper does what it says: it improves the lower bound for K(n) (the divisor function of factorial doubling) by a factor of log_3 n, bringing it to the same order as Rankin's 1938 bound for prime gaps. The main new idea is to leave some integers uncovered in the Erdős–Rankin sieve and control their contribution via averaging over a translation parameter. That is clever and it works. The paper is well-written, the estimates are standard and properly handled, and the author is transparent about the AI assistance and has independently verified the argument.

The soft spots are minor. The bound relies on de Bruijn's smooth number asymptotics at large v (log y small relative to log L), where the Dickman–de Bruijn function gives an expression like (log u)^{-1/(1-ε)+o(1)}. That is a known asymptotic, so it is fine, but one could ask whether the error terms could propagate. The author accounts for them, and the safety margin (ε/2) keeps things clean. The other potential concern is whether the constant (3/(4π^2)) is optimal—it probably isn't, but improving that is not the point.

Overall, this is a genuine advance within analytic number theory. The exposition is thorough, the logic is clear, and the proof passes muster. I would bring it to a reading group for people working on divisor functions or prime gaps. I would not cite it directly unless I worked on related sieve problems, but it is citable. A serious referee should handle it; the paper deserves publication.

Recommendation: accept after a routine check of the smooth number constant. No serious issues.

Referee report

Significance

The result is a genuine and substantial quantitative advance on the 1996 EGIP paper: the bound for K(n) moves to the classical Rankin scale, with explicit constant 3/(4π²) and an improvement factor of order log_3 n. The technical novelty is real and likely transferable: instead of covering every survivor of the Erdős–Rankin sieve, the proof leaves the residual set uncovered and controls its contribution by a first moment over a free lattice parameter, adapting the post-Rankin improvements of the prime-gap problem to the different constraint that only the cumulative S-sum matters. My line-by-line check found the constants and error terms consistent throughout: the Mertens/PNT counts in (5.1)–(5.11), the smooth-number bound (5.9) with its error terms, the o(M³) bounds (5.15) and (5.18), the Lemma 5.2 estimate with its Brun–Titchmarsh application, and the iterated-logarithm conversion (5.23) all check out, and no parameter is fitted to data.

Minor objections (6)

[§4, Lemma 4.1] In the proof of Lemma 4.1 the notation is inconsistent with its own definition: ν_p(m) was defined as the exponent of p in m!, so the displayed equation should read ν_p(n!) = ν_p((n−1)!) + a when p^a ‖ n, not ν_p(n) = ν_p(n−1) + a.
[§5, Lemma 5.1] Lemma 5.1 is quoted from [5, Lemma 2] together with a uniformity assertion ("their proof uses L only through the count of admissible t, and is therefore uniform in L over the range needed here"). Since the exact statement in [5] is not reproduced and the range ℓ ≤ u³ is specific to this paper, I recommend quoting the original lemma and justifying the uniformity, or including the short proof. This is a documentation point rather than a blocking one, since (5.18) is o(M³) and therefore insensitive to constants.
[§5, Eq. (5.18)] The left-hand side of (5.18) is restricted to covered t, while N(ℓ) counts all t ∈ [1, L], so the displayed inequality rests on an intentional overcount; a sentence stating this explicitly would prevent the impression of a gap between the two sides.
[§5, Eqs. (5.8)–(5.9)] The handling of the de Bruijn input is compressed: the uniform version of [1, (1.3), (1.4)] being invoked, the footnote justifying the omission of the additive error, and the verification that v < log y with relative error o(1) are asserted in a few lines. The claims are correct (log y = (log u)^{1+o(1)} and v²R(y) = exp(−(log u)^{1/2+o(1)}) = o(ρ(v))), but a slightly expanded derivation would make the uniformity transparent to the reader.
[§1, after (1.3)] Since 3/(4π²) ≈ 0.076 is smaller than EGIP's 1/9, the gain over (1.3) is purely asymptotic: the ratio of the new bound to the old one is (27/(4π²))(1+o(1)) log_3 n. A one-sentence remark to this effect would avoid a misreading of the constant comparison.
[§6, second session] The unverified second-session argument reported in Section 6 (claimed constant 1/300) is properly quarantined: the paper states that no correctness claim is made for it and that it is not used. This flag should remain prominent, as it currently is, so that the provenance discussion is not mistaken for a second theorem.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

This paper improves a lower bound on the number of consecutive integers needed to double n!'s divisor function, bringing it to the same order as a classical prime-gap bound.

The reading

This paper proves that for infinitely many n, the least positive integer K with d((n+K)!) ≥ 2d(n!) exceeds (3/($4π^{2}$) − ε)(log n)(log₂ n)(log₄ n)/(log₃ n)² for any ε > 0. This improves a previous lower bound by a factor of log₃ n, raising the order from (log n)(log₂ n)(log₄ n)/(log₃ n)³ to the classical scale (log n)(log₂ n)(log₄ n)/(log₃ n)². The proof adapts a sieve construction for large prime gaps to a weighted variant: instead of covering every integer in a long interval with a prime divisor from a specially built modulus, some integers are left uncovered and their contribution to the sum of prime factors is bounded on average over a translation parameter. The main technical novelty is an averaging argument that permits this relaxation.

What carries the argument

The key machinery is a weighted variant of the sieve construction used for large prime gaps. Residue classes modulo primes up to u (a parameter tending to infinity) are chosen to filter integers that have large prime factors, leaving a residual set T₂ of size roughly L log₂ u / log u. Unlike the classical approach, the integers in T₂ are not covered by additional prime divisors; instead, their contribution to the sum of prime factors S(n) is controlled on average over a range of translation parameters j via an averaging argument. The critical estimate (Lemma 5.2) uses the large sieve inequality to bound primes in arithmetic progressions arising from the uncovered integers, allowing the interval length L to be taken larger, namely L = ⌊c₀ u log u log₃ u/(log₂ u)²⌋ with c₀ = 3(1−ε)²/π².

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any ε > 0, there exist infinitely many integers n such that the least integer K with d((n+K)!) ≥ 2 d(n!) satisfies K(n) > (3/(4π²) − ε)(log n)(log₂ n)(log₄ n)/(log₃ n)². This improves the previous lower bound (1/9)(log n)(log₂ n)(log₄ n)/(log₃ n)³ established in earlier work, by a factor of order log₃ n, and matches the order of a classical lower bound for gaps between consecutive primes.

Load-bearing premise

The proof relies on the asymptotic estimate for the number of y-smooth integers up to L, specifically that Ψ(L, y) ≪ L/(log u)^{1+ε/2} under the chosen parameter relations between u, L, and y. This bound follows from known asymptotics of the smooth-number counting function (the Dickman–de Bruijn function) as a key parameter v tends to infinity; any significant deviation from the expected asymptotic could cause the bound to fail.

Editorial extensions

If this is right

The lower bound for K(n) now matches the order of the classical prime-gap lower bound, showing that the two problems are equivalent at this order.
The proof demonstrates that the weighted problem does not require full covering of the interval; controlling the average contribution of uncovered integers suffices, which separates it from the prime-gap covering problem.
Future improvements to the prime-gap covering stage do not automatically transfer to K(n) because the limiting factor is the average cost of uncovered integers, not the number of primes available for covering.
The transfer principle (Proposition 4.2) ensures that any improvement to the bound for f(n) at a slowly varying scale directly yields a corresponding improvement for K(n) with the constant halved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The averaging technique over the translation parameter j could be applied to other weighted sieve problems where leaving some positions uncovered is acceptable, potentially improving bounds for related divisor sums or additive combinatorial quantities.
The result suggests that the weighted variant of the Erdős–Rankin construction is accessible at the same depth as the classical prime-gap construction, which may inspire further refinements using more advanced tools such as the Maynard–Ford–Green–Konyagin–Tao methods modified for the weighted setting.

Original abstract

Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erd\H{o}s--Rankin construction. We follow the method of Erd\H{o}s, Graham, Ivi\'c and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,228 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.54 · human participants 0.03 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.94 · compares with prior results 0.97 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erd\H{o}s--Rankin construction. We follow the method of Erd\H{o}s, Graham, Ivi\'c and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$.	background 0.99	0.16	0.03	asserted 0.67
s1	Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3.	background 0.99	0.03	0.16	asserted 1.00
s2	\end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes.	result 0.78	0.91	0.66	asserted 1.00
s3	The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erd\H{o}s--Rankin construction.	interpretation 0.82	0.46	0.07	asserted 1.00
s4	We follow the method of Erd\H{o}s, Graham, Ivi\'c and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered.	method 0.97	0.89	0.05	asserted 1.00
s5	This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.	method 0.51	0.69	0.03	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (20)

[1]

de Bruijn, N. G. “On the number of positive integers ⩽ 𝑥 and free of prime factors > 𝑦 . ” Proc. Kon. Ned. Akad. Wetensch. Ser. A 54(1):50–60, 1951. doi:10.1016/S1385-7258(51)50008-2

View on Pith doi:10.1016/s1385-7258(51)50008-2 1951
[2]

Eberhard, S. “Ratios of consecutive values of the divisor function. ” J. Number Theory 281:426–428, 2026. doi:10.1016/j.jnt.2025.10.002

View on Pith doi:10.1016/j.jnt.2025.10.002 2026
[3]

Erdős, P. “On the difference of consecutive primes. ” Quart. J. Math. Oxford Ser. os-6(1):124–128, 1935. doi:10.1093/qmath/os-6.1.124

View on Pith doi:10.1093/qmath/os-6.1.124 1935
[4]

Erdős, P. “Some problems on number theory. ” In Proceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1986), Congr. Numer. 54:225–244,

View on Pith 1986
[5]

Erdős, P., S. W. Graham, A. Ivić and C. Pomerance. “On the number of divisors of 𝑛!. ” In Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Vol. 1 (Allerton Park, IL, 1995), B. C. Berndt, H. G. Diamond and A. J. Hildebrand (eds.), Progr. Math. 138:337–355. Birkhäuser, Boston, 1996. doi:10.1007/978-1-4612-4086-0_19

View on Pith doi:10.1007/978-1-4612-4086-0_19 1995
[6]

Erdős Problems. “Proof claims. ”Discussion thread for Erdős Problem #4, maintained by T. F. Bloom, accessed 13 September 2026; archived at web.archive.org. 22 T. FREIBERG

View on Pith 2026
[7]

Ford, K., B. Green, S. Konyagin and T. Tao. “Large gaps between consecutive prime numbers. ” Ann. of Math. (2) 183(3):935–974, 2016. doi:10.4007/annals.2016.183.3.4

View on Pith doi:10.4007/annals.2016.183.3.4 2016
[8]

Ford, K., B. Green, S. Konyagin, J. Maynard and T. Tao. “Long gaps between primes. ” J. Amer. Math. Soc. 31(1):65–105, 2018. doi:10.1090/jams/876

View on Pith doi:10.1090/jams/876 2018
[9]

Goldston, D. A., S. W. Graham, J. Pintz and C. Y. Yıldırım. “Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers. ”Int. Math. Res. Not. IMRN 2011(7):1439–1450,

View on Pith 2011
[10]

GPT 5.6 Sol. “A tilted residue-class construction for long prime-free intervals. ”Preprint, 48 pp., 25 August 2026; archived at web.archive.org. Lean 4 formalization by B. Alexeev at github.com/plby/lean-proofs

View on Pith 2026
[11]

Hasanalizade, E. “Distribution of the divisor function at consecutive integers. ” Bull. Aust. Math. Soc. 103(2):244–247, 2021. doi:10.1017/S0004972720000738

View on Pith doi:10.1017/s0004972720000738 2021
[12]

Hildebrand, A. “The divisor function at consecutive integers. ” Pacific J. Math. 129(2):307–319, 1987. doi:10.2140/pjm.1987.129.307

View on Pith doi:10.2140/pjm.1987.129.307 1987
[13]

Maynard, J. “Large gaps between primes. ” Ann. of Math. (2) 183(3):915–933, 2016. doi:10.4007/annals.2016.183.3.3

View on Pith doi:10.4007/annals.2016.183.3.3 2016
[14]

Montgomery, H. L. and R. C. Vaughan. “The large sieve. ” Mathematika 20(2):119–134, 1973. doi:10.1112/S0025579300004708

View on Pith doi:10.1112/s0025579300004708 1973
[15]

OpenAI. “Improved long gaps between primes. ” Preprint, 8 pp., 2026; archived at web.archive.org. Lean 4 formalization at github.com/openai/LongGapsBetweenPrimes. The paper attributes the proof to GPT 6 Astra

View on Pith 2026
[16]

Rankin, R. A. “The difference between consecutive prime numbers. ” J. London Math. Soc. s1-13(4):242–247,

View on Pith
[17]

Schlage-Puchta, J.-C. “On a problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers. ” Preprint, 3 pp., 2025. arXiv:2504.11463 LONG RUNS OF INTEGERS WITH SMALL PRIME FACTORS 23 Appendix A. Record of the generative session This appendix documents the generative-AI interaction described in Section 6. It is not part of the ...

View on Pith Pith review arXiv 2025
[1938]

doi:10.1112/jlms/s1-13.4.242

View on Pith doi:10.1112/jlms/s1-13.4.242
[1986]

users.renyi.hu/~p_erdos/1986-16.pdf

View on Pith 1986
[2011]

doi:10.1093/imrn/rnq124

View on Pith doi:10.1093/imrn/rnq124

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

de Bruijn, N. G. “On the number of positive integers ⩽ 𝑥 and free of prime factors > 𝑦 . ” Proc. Kon. Ned. Akad. Wetensch. Ser. A 54(1):50–60, 1951. doi:10.1016/S1385-7258(51)50008-2

View on Pith doi:10.1016/s1385-7258(51)50008-2 1951

[2] [2]

Eberhard, S. “Ratios of consecutive values of the divisor function. ” J. Number Theory 281:426–428, 2026. doi:10.1016/j.jnt.2025.10.002

View on Pith doi:10.1016/j.jnt.2025.10.002 2026

[3] [3]

Erdős, P. “On the difference of consecutive primes. ” Quart. J. Math. Oxford Ser. os-6(1):124–128, 1935. doi:10.1093/qmath/os-6.1.124

View on Pith doi:10.1093/qmath/os-6.1.124 1935

[4] [4]

Erdős, P. “Some problems on number theory. ” In Proceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1986), Congr. Numer. 54:225–244,

View on Pith 1986

[5] [5]

Erdős, P., S. W. Graham, A. Ivić and C. Pomerance. “On the number of divisors of 𝑛!. ” In Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Vol. 1 (Allerton Park, IL, 1995), B. C. Berndt, H. G. Diamond and A. J. Hildebrand (eds.), Progr. Math. 138:337–355. Birkhäuser, Boston, 1996. doi:10.1007/978-1-4612-4086-0_19

View on Pith doi:10.1007/978-1-4612-4086-0_19 1995

[6] [6]

Erdős Problems. “Proof claims. ”Discussion thread for Erdős Problem #4, maintained by T. F. Bloom, accessed 13 September 2026; archived at web.archive.org. 22 T. FREIBERG

View on Pith 2026

[7] [7]

Ford, K., B. Green, S. Konyagin and T. Tao. “Large gaps between consecutive prime numbers. ” Ann. of Math. (2) 183(3):935–974, 2016. doi:10.4007/annals.2016.183.3.4

View on Pith doi:10.4007/annals.2016.183.3.4 2016

[8] [8]

Ford, K., B. Green, S. Konyagin, J. Maynard and T. Tao. “Long gaps between primes. ” J. Amer. Math. Soc. 31(1):65–105, 2018. doi:10.1090/jams/876

View on Pith doi:10.1090/jams/876 2018

[9] [9]

Goldston, D. A., S. W. Graham, J. Pintz and C. Y. Yıldırım. “Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers. ”Int. Math. Res. Not. IMRN 2011(7):1439–1450,

View on Pith 2011

[10] [10]

GPT 5.6 Sol. “A tilted residue-class construction for long prime-free intervals. ”Preprint, 48 pp., 25 August 2026; archived at web.archive.org. Lean 4 formalization by B. Alexeev at github.com/plby/lean-proofs

View on Pith 2026

[11] [11]

Hasanalizade, E. “Distribution of the divisor function at consecutive integers. ” Bull. Aust. Math. Soc. 103(2):244–247, 2021. doi:10.1017/S0004972720000738

View on Pith doi:10.1017/s0004972720000738 2021

[12] [12]

Hildebrand, A. “The divisor function at consecutive integers. ” Pacific J. Math. 129(2):307–319, 1987. doi:10.2140/pjm.1987.129.307

View on Pith doi:10.2140/pjm.1987.129.307 1987

[13] [13]

Maynard, J. “Large gaps between primes. ” Ann. of Math. (2) 183(3):915–933, 2016. doi:10.4007/annals.2016.183.3.3

View on Pith doi:10.4007/annals.2016.183.3.3 2016

[14] [14]

Montgomery, H. L. and R. C. Vaughan. “The large sieve. ” Mathematika 20(2):119–134, 1973. doi:10.1112/S0025579300004708

View on Pith doi:10.1112/s0025579300004708 1973

[15] [15]

OpenAI. “Improved long gaps between primes. ” Preprint, 8 pp., 2026; archived at web.archive.org. Lean 4 formalization at github.com/openai/LongGapsBetweenPrimes. The paper attributes the proof to GPT 6 Astra

View on Pith 2026

[16] [16]

Rankin, R. A. “The difference between consecutive prime numbers. ” J. London Math. Soc. s1-13(4):242–247,

View on Pith

[17] [17]

Schlage-Puchta, J.-C. “On a problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers. ” Preprint, 3 pp., 2025. arXiv:2504.11463 LONG RUNS OF INTEGERS WITH SMALL PRIME FACTORS 23 Appendix A. Record of the generative session This appendix documents the generative-AI interaction described in Section 6. It is not part of the ...

View on Pith Pith review arXiv 2025

[18] [1938]

doi:10.1112/jlms/s1-13.4.242

View on Pith doi:10.1112/jlms/s1-13.4.242

[19] [1986]

users.renyi.hu/~p_erdos/1986-16.pdf

View on Pith 1986

[20] [2011]

doi:10.1093/imrn/rnq124

View on Pith doi:10.1093/imrn/rnq124