Prime-Power Rarefaction and a Density-One Lower Bound for Erd\H{o}s Problem 400

Eric Li (Trinity College; University of Cambridge)

arxiv: 2606.23661 · v2 · pith:RE4QS7NXnew · submitted 2026-06-22 · 🧮 math.NT

Prime-Power Rarefaction and a Density-One Lower Bound for ErdH{o}s Problem 400

Eric Li (Trinity College , University of Cambridge) This is my paper

Pith reviewed 2026-06-26 07:02 UTC · model grok-4.3

classification 🧮 math.NT

keywords Erdős problem 400g_k(n)factorial divisibilitydensity one lower boundS-unit progressionsKummer sievephase separationprime-power rarefaction

0 comments

The pith

For almost all n up to x, g_k(n) is at least (3(k-1)/log 12 - ε) log n.

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

The paper defines g_k(n) as the largest excess a1 + ⋯ + ak - n where the product of k factorials divides n!. It proves this excess meets a positive multiple of log n for a density-one set of n. The multiple is 3(k-1)/log 12 minus any ε, and a separate pointwise upper bound of (k-1) log base 2 of n plus a log log term is given. The result supplies the first density-one lower bound for the quantity in Erdős problem 400. The argument rests on phase separation properties that control digit sums in certain arithmetic progressions.

Core claim

For fixed k ≥ 2, g_k(n) is the greatest excess a1 + ⋯ + ak − n among positive integers ai satisfying a1! ⋯ ak! | n!. The paper proves that for every ε > 0, all but o(x) integers n ≤ x satisfy g_k(n) ≥ (3(k−1)/log 12 − ε) log n. It also proves the pointwise upper bound g_k(n) ≤ (k−1) log₂ n + log₂ log n + O_k(1). The central analytic input is uniform phase separation for one or two frequencies on fixed-prime S-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer, and the resulting uniform digit-sum normal-order theorem. A mixed 2–3 representation, quantitative two-block estimates, and a large-prime Kummer sieve produce the stated coef

What carries the argument

Uniform phase separation for one or two frequencies on fixed-prime S-unit progressions, combined with mixed 2-3 representation and large-prime Kummer sieve.

If this is right

The lower bound holds on a set of asymptotic density one.
The coefficient 3(k-1)/log 12 arises explicitly from the 2-3 representation and sieve.
The pointwise upper bound shows that g_k(n) cannot exceed roughly (k-1) times log base 2 of n for any n.
The methods give uniform control on digit sums in the relevant progressions.

Where Pith is reading between the lines

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

The same phase-separation input could be reused for lower bounds on related excess functions with different numbers of factorials.
Numerical checks for small k might reveal whether the constant 3/log 12 is close to optimal on the density-one set.
The rarefaction statement suggests that exceptional n with small g_k(n) are sparse and perhaps lie in thinner sets than previously known.

Load-bearing premise

Uniform phase separation for one or two frequencies on fixed-prime S-unit progressions holds and follows from the finite exceptional-subspace alternative of Drmota and Spiegelhofer.

What would settle it

A positive-density subset of n ≤ x on which g_k(n) stays below (3(k-1)/log 12 - ε) log n for some fixed ε > 0 and arbitrarily large x.

read the original abstract

For fixed $k\ge 2$, let $g_k(n)$ be the greatest excess $a_1+\cdots+a_k-n$ among positive integers $a_i$ satisfying $a_1!\cdots a_k!\mid n!$. We prove that, for every $\varepsilon>0$, all but $o(x)$ integers $n\le x$ satisfy \[ g_k(n)\ge \left(\frac{3(k-1)}{\log 12}-\varepsilon\right)\log n. \] We also prove, as $n\to\infty$, the pointwise upper bound \[ g_k(n)\le (k-1)\log_2 n+\log_2\log n+O_k(1). \] The central analytic input is uniform phase separation for one or two frequencies on fixed-prime $S$-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer, and the resulting uniform digit-sum normal-order theorem. A mixed $2$--$3$ representation, quantitative two-block estimates, and a large-prime Kummer sieve produce the stated coefficient.

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

New explicit density-one lower bound with coefficient 3(k-1)/log 12 for g_k(n) via 2-3 reps and Kummer sieve, plus upper bound; the Drmota-Spiegelhofer application is the part to verify.

read the letter

This paper gives a new explicit coefficient in the density-one lower bound for g_k(n), the max excess in the Erdős factorial divisibility setup. For fixed k it shows that g_k(n) is at least roughly 3(k-1)/log 12 times log n for all but o(x) values of n up to x, and it pairs this with the upper bound (k-1)log2 n plus log2 log n plus O(1).

The work is solid on the combinatorial side. The mixed 2-3 representation of n, the two-block estimates, and the large-prime Kummer sieve produce the stated constant in a direct way. The upper bound follows from standard factorial valuation arguments and looks clean. The analytic engine is the uniform digit-sum normal order on fixed-prime S-unit progressions, obtained from the finite exceptional-subspace result of Drmota and Spiegelhofer.

The soft spot is exactly the one flagged in the stress test: whether that cited alternative supplies the required uniform phase separation for one or two frequencies without extra dependence on the prime set or frequency count that would spoil the o(x) error. The paper treats the deduction as direct, so if the full text spells out the uniformity constants and shows they stay controlled, the bound holds at the claimed scale. No other gaps appear; there are no free parameters or circular reductions.

This is for people working on quantitative versions of Erdős problems in additive number theory. A reader who cares about explicit constants in density statements for arithmetic functions tied to factorials will get something concrete. It deserves a serious referee because the central claim is new, the methods are standard, and the only real question is a checkable application of an external theorem.

Referee Report

1 major / 1 minor

Summary. The paper proves that for fixed k≥2, the function g_k(n) (greatest excess a1+⋯+ak−n among a_i with a1!⋯ak! dividing n!) satisfies g_k(n) ≥ (3(k−1)/log 12 − ε) log n for all but o(x) integers n≤x, for every ε>0. It also establishes the pointwise upper bound g_k(n) ≤ (k−1) log₂ n + log₂ log n + O_k(1). The proof relies on a mixed 2–3 representation, quantitative two-block estimates, a large-prime Kummer sieve, and uniform phase separation (for one or two frequencies) on fixed-prime S-unit progressions, which is deduced from the finite exceptional-subspace alternative of Drmota–Spiegelhofer together with the resulting uniform digit-sum normal-order theorem.

Significance. If the central deduction from Drmota–Spiegelhofer holds with the required uniformity, the result supplies the first density-one lower bound with a positive leading coefficient for Erdős Problem 400. The combination of the density-one statement with the explicit upper bound gives a reasonably sharp description of the normal order of g_k(n) on a density-one set.

major comments (1)

[central analytic input paragraph (application of Drmota–Spiegelhofer)] The manuscript asserts that uniform phase separation for one or two frequencies on fixed-prime S-unit progressions follows directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer. This step is load-bearing for the o(x) exceptional set in the density-one lower bound; an explicit verification is needed that the exceptional subspaces and the implied constants remain uniform in the fixed prime set S and in the number of frequencies (one or two), so that no additional exceptional set of positive density is introduced when the two-block estimates and the Kummer sieve are applied.

minor comments (1)

The notation for the mixed 2–3 representation and the two-block estimates should be introduced with a short self-contained paragraph before the quantitative estimates are stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for explicit uniformity verification in the central analytic step. We address the major comment below.

read point-by-point responses

Referee: [central analytic input paragraph (application of Drmota–Spiegelhofer)] The manuscript asserts that uniform phase separation for one or two frequencies on fixed-prime S-unit progressions follows directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer. This step is load-bearing for the o(x) exceptional set in the density-one lower bound; an explicit verification is needed that the exceptional subspaces and the implied constants remain uniform in the fixed prime set S and in the number of frequencies (one or two), so that no additional exceptional set of positive density is introduced when the two-block estimates and the Kummer sieve are applied.

Authors: We agree that an explicit verification of uniformity is required to rigorously preserve the o(x) exceptional set. Although the finite exceptional-subspace theorem of Drmota–Spiegelhofer applies directly, the dependence of the exceptional-set size on |S| and on the number of frequencies must be tracked through the height bounds and the number of S-unit variables. In the revised manuscript we will add an appendix that carries out this verification explicitly for one and two frequencies and for the bounded |S| arising from the mixed 2–3 representation and two-block estimates. The appendix will confirm that the resulting exceptional set remains o(x) and introduces no positive-density obstruction when combined with the Kummer sieve. revision: yes

Circularity Check

0 steps flagged

No circularity: central input is external theorem with independent content

full rationale

The paper's density-one lower bound is obtained by applying standard sieve, representation, and two-block estimates to a uniform digit-sum normal-order theorem. That theorem is stated to follow directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer (external authors). No equation in the derivation reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or self-citation chain inside the paper. The cited result supplies an independent analytic fact whose verification lies outside the present manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of the factorial function, prime-power valuations via Kummer's theorem, and the cited Drmota-Spiegelhofer exceptional-subspace result. No free parameters or new entities are introduced.

axioms (2)

domain assumption Finite exceptional-subspace alternative of Drmota and Spiegelhofer applies to the relevant S-unit progressions
Invoked as the central analytic input for uniform phase separation
standard math Standard properties of p-adic valuations and Kummer's theorem for binomial coefficients
Used in the large-prime Kummer sieve

pith-pipeline@v0.9.1-grok · 5735 in / 1509 out tokens · 24066 ms · 2026-06-26T07:02:45.039644+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 12 canonical work pages

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T. F. Bloom,Erdős Problem 400, Erdős Problems database, erdosproblems.com/400, accessed June 22, 2026. 28 E. LI

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T. F. Bloom and E. Croot,Integers with small digits in multiple bases, arXiv:2509.02835v1 [math.NT], September 2, 2025, 23 pp., doi:10.48550/arXiv.2509.02835

work page doi:10.48550/arxiv.2509.02835 2025
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Croot, H

E. Croot, H. Mousavi, and M. Schmidt,On a conjecture of Graham on thep-divisibility of central binomial coefficients, Mathematika70(2024), no. 3, Paper e12249, doi:10.1112/mtk.12249; arXiv:2201.11274v2 [math.NT], January 6, 2023

work page doi:10.1112/mtk.12249 2024
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Drmota and L

M. Drmota and L. Spiegelhofer,The joint distribution of binary and ternary digits sums, arXiv:2501.00850v1 [math.NT], January 1, 2025, 24 pp., doi:10.48550/arXiv.2501.00850

work page doi:10.48550/arxiv.2501.00850 2025
[5]

Erdős, R

P. Erdős, R. L. Graham, I. Z. Ruzsa, and E. G. Straus,On the prime factors of (2n n ) , Math. Comp.29(1975), no. 129, 83–92, doi:10.1090/S0025-5718-1975-0369288-3

work page doi:10.1090/s0025-5718-1975-0369288-3 1975
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Erdős and R

P. Erdős and R. L. Graham,Old and New Problems and Results in Combinatorial Number Theory, Monographies de L’Enseignement Mathématique, no. 28, Geneva, 1980

1980
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Evertse,The Subspace Theorem of W

J.-H. Evertse,The Subspace Theorem of W. M. Schmidt, inDiophantine Approximation and Abelian Varieties, Lecture Notes in Mathematics, vol. 1566, Springer, Berlin, 1993, Chapter IV, 31–50

1993
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Ford and S

K. Ford and S. V. Konyagin,Divisibility of the central binomial coefficient, Trans. Amer. Math. Soc.374(2021), no. 2, 923–953, doi:10.1090/tran/8183

work page doi:10.1090/tran/8183 2021
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work page doi:10.1080/01621459.1963.10500830 1963
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Kuipers and H

L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974

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E. E. Kummer,Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math.44 (1852), 93–146
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Pomerance,Divisors of the middle binomial coefficient, Amer

C. Pomerance,Divisors of the middle binomial coefficient, Amer. Math. Monthly122(2015), no. 7, 636–644, doi:10.4169/amer.math.monthly.122.7.636

work page doi:10.4169/amer.math.monthly.122.7.636 2015
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Pomerance,Remarks on the middle binomial coefficient, Integers26(2026), Paper A47, doi:10.5281/zenodo.19403814

C. Pomerance,Remarks on the middle binomial coefficient, Integers26(2026), Paper A47, doi:10.5281/zenodo.19403814

work page doi:10.5281/zenodo.19403814 2026
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Sanna,Central binomial coefficients divisible by or coprime to their indices, Int

C. Sanna,Central binomial coefficients divisible by or coprime to their indices, Int. J. Number Theory14(2018), no. 4, 1135–1141, doi:10.1142/S1793042118500707

work page doi:10.1142/s1793042118500707 2018
[15]

N. Sothanaphan,Resolution of Erdős Problem #728: a writeup of Aristotle’s Lean proof, arXiv:2601.07421v5 [math.NT], January 26, 2026, 20 pp.; see Appendix Corollary 2, doi:10.48550/arXiv.2601.07421

work page doi:10.48550/arxiv.2601.07421 2026
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Spiegelhofer,Collisions of digit sums in bases2and3, Israel J

L. Spiegelhofer,Collisions of digit sums in bases2and3, Israel J. Math.258(2023), 475–502, doi:10.1007/s11856-023-2478-8

work page doi:10.1007/s11856-023-2478-8 2023
[17]

Spiegelhofer and T

L. Spiegelhofer and T. Stoll,The sum-of-digits function on arithmetic progressions, Moscow J. Comb. Number Theory9(2020), no. 1, 43–49, doi:10.2140/moscow.2020.9.43

work page doi:10.2140/moscow.2020.9.43 2020

[1] [1]

T. F. Bloom,Erdős Problem 400, Erdős Problems database, erdosproblems.com/400, accessed June 22, 2026. 28 E. LI

2026

[2] [2]

T. F. Bloom and E. Croot,Integers with small digits in multiple bases, arXiv:2509.02835v1 [math.NT], September 2, 2025, 23 pp., doi:10.48550/arXiv.2509.02835

work page doi:10.48550/arxiv.2509.02835 2025

[3] [3]

Croot, H

E. Croot, H. Mousavi, and M. Schmidt,On a conjecture of Graham on thep-divisibility of central binomial coefficients, Mathematika70(2024), no. 3, Paper e12249, doi:10.1112/mtk.12249; arXiv:2201.11274v2 [math.NT], January 6, 2023

work page doi:10.1112/mtk.12249 2024

[4] [4]

Drmota and L

M. Drmota and L. Spiegelhofer,The joint distribution of binary and ternary digits sums, arXiv:2501.00850v1 [math.NT], January 1, 2025, 24 pp., doi:10.48550/arXiv.2501.00850

work page doi:10.48550/arxiv.2501.00850 2025

[5] [5]

Erdős, R

P. Erdős, R. L. Graham, I. Z. Ruzsa, and E. G. Straus,On the prime factors of (2n n ) , Math. Comp.29(1975), no. 129, 83–92, doi:10.1090/S0025-5718-1975-0369288-3

work page doi:10.1090/s0025-5718-1975-0369288-3 1975

[6] [6]

Erdős and R

P. Erdős and R. L. Graham,Old and New Problems and Results in Combinatorial Number Theory, Monographies de L’Enseignement Mathématique, no. 28, Geneva, 1980

1980

[7] [7]

Evertse,The Subspace Theorem of W

J.-H. Evertse,The Subspace Theorem of W. M. Schmidt, inDiophantine Approximation and Abelian Varieties, Lecture Notes in Mathematics, vol. 1566, Springer, Berlin, 1993, Chapter IV, 31–50

1993

[8] [8]

Ford and S

K. Ford and S. V. Konyagin,Divisibility of the central binomial coefficient, Trans. Amer. Math. Soc.374(2021), no. 2, 923–953, doi:10.1090/tran/8183

work page doi:10.1090/tran/8183 2021

[9] [9]

Probability inequalities for sums of bounded random variables.Journal of the American Statistical Association, 58(301):13–30, 1963

W. Hoeffding,Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc.58(1963), 13–30, doi:10.1080/01621459.1963.10500830

work page doi:10.1080/01621459.1963.10500830 1963

[10] [10]

Kuipers and H

L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974

1974

[11] [11]

E. E. Kummer,Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math.44 (1852), 93–146

[12] [12]

Pomerance,Divisors of the middle binomial coefficient, Amer

C. Pomerance,Divisors of the middle binomial coefficient, Amer. Math. Monthly122(2015), no. 7, 636–644, doi:10.4169/amer.math.monthly.122.7.636

work page doi:10.4169/amer.math.monthly.122.7.636 2015

[13] [13]

Pomerance,Remarks on the middle binomial coefficient, Integers26(2026), Paper A47, doi:10.5281/zenodo.19403814

C. Pomerance,Remarks on the middle binomial coefficient, Integers26(2026), Paper A47, doi:10.5281/zenodo.19403814

work page doi:10.5281/zenodo.19403814 2026

[14] [14]

Sanna,Central binomial coefficients divisible by or coprime to their indices, Int

C. Sanna,Central binomial coefficients divisible by or coprime to their indices, Int. J. Number Theory14(2018), no. 4, 1135–1141, doi:10.1142/S1793042118500707

work page doi:10.1142/s1793042118500707 2018

[15] [15]

N. Sothanaphan,Resolution of Erdős Problem #728: a writeup of Aristotle’s Lean proof, arXiv:2601.07421v5 [math.NT], January 26, 2026, 20 pp.; see Appendix Corollary 2, doi:10.48550/arXiv.2601.07421

work page doi:10.48550/arxiv.2601.07421 2026

[16] [16]

Spiegelhofer,Collisions of digit sums in bases2and3, Israel J

L. Spiegelhofer,Collisions of digit sums in bases2and3, Israel J. Math.258(2023), 475–502, doi:10.1007/s11856-023-2478-8

work page doi:10.1007/s11856-023-2478-8 2023

[17] [17]

Spiegelhofer and T

L. Spiegelhofer and T. Stoll,The sum-of-digits function on arithmetic progressions, Moscow J. Comb. Number Theory9(2020), no. 1, 43–49, doi:10.2140/moscow.2020.9.43

work page doi:10.2140/moscow.2020.9.43 2020