Erd\H{o}s Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations

Eric Li (Trinity College; United Kingdom); University of Cambridge

arxiv: 2606.08216 · v1 · pith:N732Z3JMnew · submitted 2026-06-06 · 🧮 math.NT · math.PR

ErdH{o}s Problem 684 at Density One: Small-prime Parts of Binomial Coefficients and Gaussian Fluctuations

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

Pith reviewed 2026-06-27 19:13 UTC · model grok-4.3

classification 🧮 math.NT math.PR

keywords binomial coefficientsErdős problemsnormal orderKummer's theoremGaussian fluctuationssmall prime factorsdensity one

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The pith

The smallest k such that the small-prime part of binom(n,k) exceeds n^c equals (c/(1-γ) + o(1)) log n for almost all n.

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

The paper establishes a density-one normal-order result for the function f_c(n), defined as the least k where u(n,k) exceeds n^c, with u(n,k) the largest divisor of binom(n,k) using only primes at most k. For every fixed c greater than zero this yields f_c(n) equal to (c divided by one minus gamma plus little-o of one) times log n on a set of density one. The argument begins from Kummer's theorem, which converts log u(n,k) into a sum of base-p carry indicators, then averages those indicators over complete residue systems to obtain the mean m(k) approximately equal to (1 minus gamma) times k. The same averaging produces a Gaussian limit law for the centered and scaled log u(n,k) when k tends to infinity but remains at most A log X. A reader would care because the result supplies the typical size of this arithmetic threshold rather than only its worst-case behavior.

Core claim

For each fixed c greater than zero, f_c(n) equals (c divided by one minus gamma plus little-o of one) times log n for almost all positive integers n. In the special case c equals two this gives f_2(n) equal to (2 divided by one minus gamma plus little-o of one) times log n, or about 4.7305 log n, for the Erdős problem 684 threshold. The mean m(k) arises from complete-residue averaging of the carry indicators supplied by Kummer's theorem and equals (1 minus gamma)k plus little-o of k; after discarding a zero-density exceptional set caused by high powers of small primes dividing nearby integers, log u(n,k) concentrates around this mean uniformly for all k up to A log X. When k equals k(X) tend

What carries the argument

Kummer's theorem expressing log u(n,k) as a sum of carry indicators across primes p at most k, averaged over complete residue classes to produce the explicit mean m(k) equal to (1 minus gamma)k plus little-o of k.

If this is right

The typical crossing scale for u(n,k) greater than n^c shifts from the naive c log n to c divided by one minus gamma times log n because of the factor (1 minus gamma) in the averaged carry sum.
Gaussian fluctuations of log u(n,k) hold with explicit variance asymptotic to (2 minus log of 2 pi) k log k whenever k tends to infinity but stays below A log X.
Prime powers higher than the first contribute to the mean m(k) but become negligible in L squared norm on the scale of the Gaussian fluctuations.

Where Pith is reading between the lines

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

The same carry-averaging technique might produce normal-order statements for other functions built from binomial coefficients or from Kummer-type carry counts.
The constant one minus gamma, arising from the sum over primes of log p divided by p minus one, is likely to appear in related problems that average additive functions over residue classes.
Numerical checks on intervals [X,2X) for moderate X could verify whether the proportion of n satisfying the asymptotic for f_c(n) approaches one as predicted.

Load-bearing premise

The integers n for which a high power of some small prime divides one of n, n-1, up to a short interval around n form a set of density zero.

What would settle it

Existence of a positive-density subset of n in [X,2X) on which log u(n,k) deviates from m(k) by more than twice the square root of V(k) for some k near (c divided by one minus gamma) log n would falsify the claimed concentration.

read the original abstract

For $0\leq k\leq n$, let $u(n,k)$ be the largest divisor of $\binom nk$ whose prime factors are at most $k$. Erd\H{o}s Problem #684 concerns the special threshold $u(n,k)>n^2$ and asks how early this small-prime part can be forced to become large. We prove the density-one analogue for every fixed power threshold. If $f_c(n)$ is the least $k$ for which $u(n,k)>n^c$, then, for each fixed $c>0$, \[ f_c(n)=\left(\frac{c}{1-\gamma}+o(1)\right)\log n \] for almost all positive integers $n$. In particular, \[ f_2(n)=\left(\frac{2}{1-\gamma}+o(1)\right)\log n =(4.730544237\ldots+o(1))\log n \] for the Erd\H{o}s #684 threshold. This is a normal-order theorem, not a pointwise resolution of the corresponding worst-case problem. The constant $1-\gamma$ is arithmetic. Kummer's theorem rewrites $\log u(n,k)$ as a sum of carry indicators, and complete-residue averaging gives \[ m(k)=k\sum_{p\leq k}\frac{\log p}{p-1}-\log k!=(1-\gamma)k+o(k). \] The cancellation in this formula moves the typical crossing from the naive scale $c\log n$ to $c(1-\gamma)^{-1}\log n$. We prove the required concentration uniformly for every $k\leq A\log X$ on one dyadic interval, after discarding a zero-density exceptional set caused by large powers of small primes dividing one of the nearby integers $n,n-1,\ldots$. We also prove Gaussian fluctuations in the logarithmic range. If $k=k(X)\to\infty$, $k\leq A\log X$, and $n$ is uniform in $[X,2X)\cap\mathbb Z$, then \[ \frac{\log u(n,k)-m(k)}{\sqrt{V(k)}}\Rightarrow \mathcal N(0,1), \qquad V(k)\sim (2-\log(2\pi))k\log k. \] Higher prime powers are needed for the mean, but after centering their aggregate is $L^2$-negligible on the Gaussian scale; the variance comes only from the prime levels.

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

Paper gives the density-one normal order for Erdős problem 684 as (c/(1-γ)+o(1))log n with Gaussian fluctuations of variance ~ (2-log(2π))k log k.

read the letter

The main takeaway is that this paper settles the typical size of the smallest k where the small-prime part u(n,k) of binom(n,k) exceeds n^c. For almost all n, that threshold f_c(n) is asymptotically c/(1-γ) log n, and they also obtain Gaussian fluctuations around the mean m(k) with the stated variance.

The derivation of the mean is clean. Kummer's theorem turns log u(n,k) into a sum of carry indicators, and complete-residue averaging produces m(k) = k sum_{p≤k} (log p)/(p-1) - log k! which simplifies to (1-γ)k + o(k). That factor 1-γ moves the crossing point away from the naive c log n scale, and the constant is explicit enough to give the numerical value 4.73... for the original Erdős threshold c=2.

The concentration step removes a zero-density exceptional set of n where high powers of small primes divide one of n, n-1, … . The stress-test note correctly flags that a plain union bound over p ≤ A log X, O(log X) positions, and α ≥ 2 does not immediately yield density o(1). The paper must therefore use a truncation on α or control overlaps to make the set small uniformly in the dyadic interval. If that estimate is carried out properly, the density-one statement follows; otherwise the claim only holds after removing a larger set.

The Gaussian result is obtained by showing that higher prime powers contribute negligibly in L^2 after centering, so the variance comes from the first-power terms. This is a standard second-moment argument once the mean is in hand.

The work is for number theorists who care about normal orders in combinatorial number theory and explicit constants in Erdős-type problems. It is not a pointwise resolution, which remains open. The approach is grounded in Kummer and averaging without circularity or free parameters.

I would bring the explicit constant and the variance formula to a reading group. The paper deserves a serious referee because the claims are precise, the tools are standard, and the only real question is whether the exceptional-set density is handled rigorously.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a density-one normal-order result for Erdős problem 684: for each fixed c>0, f_c(n)=(c/(1-γ)+o(1))log n for almost all n, where f_c(n) is the least k such that the largest k-smooth divisor u(n,k) of binom(n,k) exceeds n^c. In particular f_2(n)=(2/(1-γ)+o(1))log n. The mean m(k)=(1-γ)k+o(k) is obtained from Kummer's theorem by rewriting log u(n,k) as a sum of carry indicators and averaging over complete residues. Concentration of log u(n,k) around m(k) is proved uniformly for k≤A log X on dyadic intervals after removing a zero-density exceptional set of n divisible by high powers of small primes; Gaussian fluctuations are also established with V(k)∼(2-log(2π))k log k.

Significance. If the result holds, it supplies the first density-one resolution of Erdős problem 684 together with an explicit arithmetic constant arising from Kummer averaging, shifting the threshold from the naive c log n scale. The Gaussian limit theorem is a substantial strengthening that describes the distribution on the logarithmic range. The derivation is parameter-free, uses only standard tools of analytic number theory, and yields falsifiable predictions for the normal order.

major comments (2)

[Abstract, concentration paragraph] Abstract (concentration paragraph) and the section establishing the zero-density exceptional set: the claim that E_X has density o(1) uniformly in X is load-bearing for the density-one statement. The union bound over p≤A log X, O(log X) positions, and α≥2 does not immediately yield o(1) for moderate α; a refined truncation of α depending on p together with overlap control is required. The manuscript asserts the estimate exists, but the specific bound (e.g., the contribution of p^α with α=2,3,… up to log log X) must be exhibited to confirm uniformity.
[Gaussian fluctuations paragraph] The Gaussian-fluctuation statement: after centering, higher prime powers are asserted to be L^2-negligible on the scale sqrt(V(k)), with variance arising only from the prime levels. The precise L^2 estimate separating the prime-level contribution from the higher-power tail should be checked against the variance formula V(k)∼(2-log(2π))k log k.

minor comments (2)

Notation: the definition of u(n,k) as the largest divisor of binom(n,k) with primes ≤k is clear, but the range of k in the uniform concentration statement (k≤A log X) should be restated explicitly when the exceptional set is introduced.
The numerical constant 4.730544237… for 2/(1-γ) is given to nine decimals; a brief reference or computation note for γ would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the need for explicit verification of the zero-density bound and the L^2 separation in the Gaussian regime. Both points are addressable by adding detailed calculations that were omitted for brevity; we will incorporate them in the revision.

read point-by-point responses

Referee: [Abstract, concentration paragraph] Abstract (concentration paragraph) and the section establishing the zero-density exceptional set: the claim that E_X has density o(1) uniformly in X is load-bearing for the density-one statement. The union bound over p≤A log X, O(log X) positions, and α≥2 does not immediately yield o(1) for moderate α; a refined truncation of α depending on p together with overlap control is required. The manuscript asserts the estimate exists, but the specific bound (e.g., the contribution of p^α with α=2,3,… up to log log X) must be exhibited to confirm uniformity.

Authors: We agree that an explicit computation of the density of E_X is required for uniformity in X. In the revised manuscript we will insert a dedicated lemma (new Lemma 3.4) that performs the refined truncation: for each prime p ≤ A log X we sum only up to α ≤ (log log X)/log p + O(1), bound the tail by geometric series, and control overlaps via the fact that at most O(log X) consecutive integers are considered. The resulting total measure is ≪ (log X) ∑_{p≤A log X} ∑_{α≥2, α≤(log log X)/log p} X^{-1} p^{-α} + o(1) which evaluates to o(1) uniformly, with the α=2,3 terms displayed explicitly and shown to be O(1/log log X). revision: yes
Referee: [Gaussian fluctuations paragraph] The Gaussian-fluctuation statement: after centering, higher prime powers are asserted to be L^2-negligible on the scale sqrt(V(k)), with variance arising only from the prime levels. The precise L^2 estimate separating the prime-level contribution from the higher-power tail should be checked against the variance formula V(k)∼(2-log(2π))k log k.

Authors: The L^2 negligibility is proved in Section 5 by writing Var(log u(n,k)) = Var(sum_p I_p) + 2 Cov + Var(higher powers), where the higher-power term is bounded by O(k) via the same Kummer carry indicators for α≥2. Since V(k) ∼ (2-log(2π))k log k, the ratio O(k)/V(k) = O(1/log k) → 0, so the centered higher-power contribution is o_p(sqrt(V(k))). We will add an explicit comparison paragraph immediately after the variance asymptotic, displaying the O(k) bound next to the leading term of V(k) to make the separation transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses external theorems and estimates

full rationale

The paper derives m(k) explicitly from Kummer's theorem (rewriting log u(n,k) as carry indicators) followed by complete-residue averaging, which produces the arithmetic constant 1-γ via standard prime-sum asymptotics; this is an independent calculation, not a fit or self-definition. The density-one statement follows from a uniform concentration argument after removing a zero-density exceptional set E_X whose size is bounded by direct estimates on high powers p^α dividing nearby integers (no reduction of the target f_c(n) asymptotic to any fitted input). Gaussian fluctuations are obtained from an L^2 variance computation over prime levels, again independent of the main claim. No self-citations, ansatzes, or renamings appear as load-bearing steps. The derivation chain is therefore self-contained against external number-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on Kummer's theorem (standard) and complete-residue averaging (standard analytic number theory); no free parameters are fitted to the target result and no new entities are introduced.

axioms (1)

standard math Kummer's theorem rewrites log u(n,k) as a sum of carry indicators
Invoked in the abstract to express the small-prime part.

pith-pipeline@v0.9.1-grok · 6008 in / 1380 out tokens · 27554 ms · 2026-06-27T19:13:08.732137+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references

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[1] [1]

Alexeev, M

B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant,Short proofs in combinatorics and number theory, arXiv:2603.29961v2 [math.CO], 2026

arXiv 2026

[2] [2]

Billingsley,Probability and Measure, 3rd ed., Wiley, 1995

P. Billingsley,Probability and Measure, 3rd ed., Wiley, 1995

1995

[3] [3]

T. F. Bloom,Erdős Problem #684, Erdős Problems.https://www.erdosproblems.com/684, accessed 6 June 2026

2026

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Erdős,Some unconventional problems in number theory, Acta Mathematica Academiae Scientiarum Hungaricae 33(1979), 71–80

P. Erdős,Some unconventional problems in number theory, Acta Mathematica Academiae Scientiarum Hungaricae 33(1979), 71–80

1979

[5] [5]

E. E. Kummer,Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik44(1852), 93–146

[6] [6]

H. L. Montgomery and R. C. Vaughan,Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2007

2007

[7] [7]

H. P. Rosenthal,On the subspaces ofLp (p> 2) spanned by sequences of independent random variables, Israel Journal of Mathematics8(1970), 273–303

1970