Decomposing a factorial into large factors

Andrew V. Sutherland; Boris Alexeev; Evan Conway; Kevin Ventullo; Markus Uhr; Matthieu Rosenfeld; Terence Tao

arxiv: 2503.20170 · v4 · submitted 2025-03-26 · 🧮 math.NT

Decomposing a factorial into large factors

Boris Alexeev , Evan Conway , Matthieu Rosenfeld , Andrew V. Sutherland , Terence Tao , Markus Uhr , Kevin Ventullo This is my paper

Pith reviewed 2026-05-22 23:22 UTC · model grok-4.3

classification 🧮 math.NT

keywords factorial decompositiont(N) sequenceasymptotic expansionErdős-Graham problemproduct of integersnumber theoryexplicit boundsGuy-Selfridge conjectures

0 comments

The pith

t(N) satisfies t(N)/N = 1/e - c0/log N + O(1/log^{1+c} N) with explicit c0 = 0.30441901…

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

The paper studies t(N), the largest integer such that N! equals a product of exactly N integers each at least t(N). It establishes the refined asymptotic formula t(N)/N = 1/e − c0/log N + O(1/log^{1+c} N) for an explicit constant c0 ≈ 0.30441901 and some c > 0, sharpening an earlier unpublished result of Erdős, Selfridge, and Straus that only gave t(N)/N = 1/e − o(1). The same work supplies a further lower-order term in the upper bound and uses extensive computation to prove that t(N) ≥ N/3 holds for all N ≥ 43632, with the threshold shown to be optimal. These statements directly answer a question posed by Erdős and Graham and settle several explicit conjectures of Guy and Selfridge.

Core claim

The central claim is that the ratio t(N)/N admits the expansion 1/e − c0/log N + O(1/log^{1+c} N) with explicit numerical c0 = 0.30441901…, together with the concrete inequality t(N) ≥ N/3 for every N ≥ 43632 that is best possible. The proof of the asymptotic supplies an additional lower-order term on the upper-bound side; the explicit lower bound is obtained by a combination of analytic estimates and high-precision numerical verification over wide ranges of N.

What carries the argument

t(N), defined as the largest integer such that N! equals a product of N integers each ≥ t(N)

If this is right

t(N)/N approaches 1/e from below at the explicit rate c0/log N
t(N) is at least N/3 once N reaches 43632, and 43632 is the smallest such threshold
A second lower-order term appears in the upper bound for t(N)/N
Several explicit conjectures of Guy and Selfridge on the sequence t(N) are confirmed by the computed values

Where Pith is reading between the lines

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

The same decomposition question can be posed for other highly composite sequences such as primorials or lcm[1..N]
The explicit constant c0 may be expressible in closed form involving integrals or series that arise from optimizing the product under the factorial constraint
The computational method used to verify the N/3 threshold could be adapted to locate the exact point where t(N) first exceeds N/k for other small k

Load-bearing premise

The analytic and combinatorial arguments that produce the explicit value of c0 and the stated error term are correct and complete.

What would settle it

A direct computation of t(N) for some N > 10^5 that lies outside the predicted interval N/e − c0 N/log N − C N/log^{1+c} N < t(N) < N/e − c0 N/log N + C N/log^{1+c} N for any fixed C would falsify the asymptotic.

read the original abstract

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erd\H{o}s and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

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

Explicit c0 and computational resolution of the N/3 threshold for t(N).

read the letter

The paper gives the first explicit value for the constant c0 in the expansion t(N)/N = 1/e - c0 / log N + O(1/log^{1+c} N), answering the Erdős-Graham question directly. It also uses heavy computation to confirm that t(N) >= N/3 for all N at least 43632, with the bound sharp. The new part is the concrete c0 = 0.30441901... and the additional lower order term on the upper bound. The numerical work settles multiple Guy-Selfridge conjectures with high precision. The analytic derivation appears solid from the abstract, and the explicit constant is a genuine advance over the o(1) result. The computations are presented as independent and precise. The soft spot is the non-explicit c in the error term. At N=43632 the difference between 1/e - c0/log N and 1/3 is small, about 0.006, so the O term needs to be controlled to justify the inequality without computation. The paper invokes numerical assistance for wide ranges, which likely means they computed t(N) directly for N up to well beyond where the error is negligible. If that range is documented and the computations are rigorous, the claim holds; if not, there's a minor issue with the justification for the exact threshold. Overall the work is careful. It is aimed at number theorists interested in precise asymptotics and conjectures on factorial factorizations. Anyone following Erdős problems on this topic will find it useful. I would send it to peer review. The explicit constant and the computational resolutions make it worth the referees' time.

Referee Report

2 major / 0 minor

Summary. The paper defines t(N) as the largest integer such that N! factors as a product of exactly N integers each at least t(N). It establishes the refined asymptotic t(N)/N = 1/e - c0/log N + O(1/log^{1+c} N) with explicit c0 = 0.30441901…, together with a further lower-order term in the upper bound. Using numerical assistance, it reports highly precise values of t(N) over wide ranges and proves the explicit statement that t(N) ≥ N/3 for all N ≥ 43632, with the threshold shown to be best possible; several conjectures of Guy and Selfridge are thereby settled.

Significance. If the results hold, the work supplies the first explicit constant in the expansion of t(N)/N, recovers and sharpens the lost Erdős–Selfridge–Straus result, and answers a question of Erdős and Graham. The combination of an explicit leading correction term with a sharp finite-N threshold verified by computation is a concrete advance in combinatorial number theory.

major comments (2)

[asymptotic expansion (statement of main theorem)] The error term is written O(1/log^{1+c} N) for an unspecified absolute constant c > 0. At N = 43632 the difference between the main terms 1/e - c0/log N and 1/3 is only ≈ 0.006; without an effective bound on the implied constant in the O-term, the asymptotic alone does not rigorously imply t(N) ≥ N/3 at this finite threshold. The manuscript must therefore supply either an effective version of the error or a fully explicit bridging argument that combines the asymptotic with the cited numerical assistance.
[numerical results and explicit threshold] The explicit threshold claim t(N) ≥ N/3 for N ≥ 43632 (and its sharpness) rests on “numerical assistance” and “highly precise computations.” No description is given of the computational method, the range of direct verification versus asymptotic tail, the precision or error analysis of the computations, or the auxiliary bounds used to certify the inequality at the cutoff. These details are load-bearing for the finite-N statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which highlight the need for greater rigor and transparency in establishing the finite-N threshold. We address each major comment below.

read point-by-point responses

Referee: [asymptotic expansion (statement of main theorem)] The error term is written O(1/log^{1+c} N) for an unspecified absolute constant c > 0. At N = 43632 the difference between the main terms 1/e - c0/log N and 1/3 is only ≈ 0.006; without an effective bound on the implied constant in the O-term, the asymptotic alone does not rigorously imply t(N) ≥ N/3 at this finite threshold. The manuscript must therefore supply either an effective version of the error or a fully explicit bridging argument that combines the asymptotic with the cited numerical assistance.

Authors: We agree that the non-effective error term prevents the asymptotic from directly implying the threshold at N=43632. The claimed inequality t(N) ≥ N/3 for N ≥ 43632 is established by a combination of direct computation up to a suitable cutoff and the asymptotic for larger N. In the revision we will supply either an effective version of the O-term (by determining an explicit c and bounding the implied constant) or a fully explicit bridging argument that uses the numerical data to cover the transitional range. This will be incorporated into the main theorem statement and its proof. revision: yes
Referee: [numerical results and explicit threshold] The explicit threshold claim t(N) ≥ N/3 for N ≥ 43632 (and its sharpness) rests on “numerical assistance” and “highly precise computations.” No description is given of the computational method, the range of direct verification versus asymptotic tail, the precision or error analysis of the computations, or the auxiliary bounds used to certify the inequality at the cutoff. These details are load-bearing for the finite-N statement.

Authors: We acknowledge that the manuscript provides insufficient detail on the computational verification. In the revised version we will add a dedicated section (or appendix) describing the algorithm for computing t(N), the precise range of direct verification (including up to and beyond N=43632), the floating-point precision employed, error analysis and certification bounds, and any auxiliary inequalities used to confirm the threshold. This will make the finite-N claims fully rigorous and reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and bounds are self-contained

full rationale

The paper derives the refined asymptotic t(N)/N = 1/e - c0/log N + O(1/log^{1+c} N) with explicit c0 from its own analytic arguments, independent of the lost Erdős-Selfridge-Straus result (which is cited only for historical context on the o(1) term). The explicit finite-N claim t(N) ≥ N/3 for N ≥ 43632 rests on separate numerical computations described as 'highly precise' and 'with numerical assistance,' not on fitting or renaming the asymptotic itself. No self-citation is load-bearing for the central results, no parameter is fitted to a subset and then called a prediction, and no ansatz or uniqueness theorem is smuggled via prior author work. The non-explicit c in the error term affects effectivity but does not induce circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, axioms, or invented entities; the constant c0 is stated as explicit but its derivation method is not shown.

pith-pipeline@v0.9.0 · 5768 in / 1242 out tokens · 74332 ms · 2026-05-22T23:22:31.626019+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

t(N)/N = 1/e − c0/log N + O(1/log^{1+c} N) with c0 = (1/e) ∫ f_e(x) dx
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

rearrangement of powers of 2,3,5,7; downsets; 3-smooth liquidity pool

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · cited by 1 Pith paper

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[1] [1]

Alladi, C

K. Alladi, C. Grinstead,On the decomposition of𝑛!into prime powers, J. Number Theory9(1977) 452– 458

work page 1977

[2] [2]

Algorithms5(1984) 502–525

S.F.Assmann,D.S.Johnson,D.J.Kleitman,J.Y.-T.Leung,Onadualversionoftheone-dimensionalbin packing problem, J. Algorithms5(1984) 502–525

work page 1984

[3] [3]

Baker, G

A. Baker, G. Wüstholz,Logarithmic forms and Diophantine geometry, New Math. Monogr., 9 Cambridge University Press, Cambridge, 2007

work page 2007

[4] [4]

Büthe,Estimating𝜋(𝑥)and related functions under partial RH assumptions, Math

J. Büthe,Estimating𝜋(𝑥)and related functions under partial RH assumptions, Math. Comp., 85(301), 2483–2498, Jan. 2016

work page 2016

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Büthe,An analytic method for bounding𝜓(𝑥)

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work page 1991

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Rivat,Computing𝜋(𝑥): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math

M.Deléglise, J. Rivat,Computing𝜋(𝑥): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp.65(1996), no. 213, 235–245

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P. Dusart,Explicit estimates of some functions over primes, Ramanujan J.45(2018) 227–251

work page 2018

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Erdős,Some problems in number theory, in Computers in Number Theory, Academic Press, London New York, 1971, pp

P. Erdős,Some problems in number theory, in Computers in Number Theory, Academic Press, London New York, 1971, pp. 405–414

work page 1971

[9] [9]

Erdős,Some problems I presented or planned to present in my short talk, Analytic number theory, Vol

P. Erdős,Some problems I presented or planned to present in my short talk, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333–335

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Erdős, R

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R. K. Guy,Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004

work page 2004

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work page 1998

[14] [14]

Johnston, A

D. Johnston, A. Yang,Some explicit estimates for the error term in the prime number theorem, J. Math. Anal. Appl.,527(2) (2023), Paper No. 127460

work page 2023

[15] [15]

J. C. Lagarias and A. M. Odlyzko,Computing𝜋(𝑥): An analytic method, J. Algorithms8(1987), no. 2, 173–191

work page 1987

[16] [16]

170, 537–560

J.C.LagariasandA.M.Odlyzko,Computing𝜋(𝑥): TheMiessel–Lehmermethod,Math.Comp.44(1985), no. 170, 537–560

work page 1985

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Berkelaar, K

M. Berkelaar, K. Eikland, P.Notebaert,lp_solve, Version 5.5.2.11,https://lpsolve.sourceforge. net/5.5/, accessed May 5, 2025

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Platt, T

D. Platt, T. Trudgian,The Riemann hypothesis is true up to3⋅1012, Bull. Lond. Math. Soc.53(2021), no. 3, 792–797

work page 2021

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Robbins,A Remark on Stirling’s Formula, Amer

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work page 1955

[20] [20]

Rosser, L

J. Rosser, L. Schoenfeld,Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94

work page 1962

[21] [21]

Tao,Decomposing factorials into bounded factors, preprint, 2025.https://arxiv.org/abs/2503

T. Tao,Decomposing factorials into bounded factors, preprint, 2025.https://arxiv.org/abs/2503. 20170v2

work page 2025

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Tao,Verifying the Guy–Selfridge conjecture, GitHub repository, 2025.https://github.com/ teorth/erdos-guy-selfridge

T. Tao,Verifying the Guy–Selfridge conjecture, GitHub repository, 2025.https://github.com/ teorth/erdos-guy-selfridge

work page 2025

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J. D. Vaaler,Some extremal functions in Fourier analysis, Bulletin (New Series) of the American Mathe- matical Society, Bull. Amer. Math. Soc. (N.S.)12(2), 183–216, (April 1985)

work page 1985

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Walisch,primecount, Version 7.17,https://github.com/kimwalisch/primecount, accessed April 24, 2025

K. Walisch,primecount, Version 7.17,https://github.com/kimwalisch/primecount, accessed April 24, 2025

work page 2025

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Walisch,primesieve, Version 12.8,https://github.com/kimwalisch/primesieve, accessed April 24, 2025

K. Walisch,primesieve, Version 12.8,https://github.com/kimwalisch/primesieve, accessed April 24, 2025. DECOMPOSING A FACTORIAL INTO LARGE FACTORS 79 UNAFFILIATED, ATHENS, GA 30605. Email address:boris.alexeev@gmail.com UVA DEPARTMENT OFMATHEMATICS, CHARLOTTESVILLE, VA 22903. Email address:auj4kq@virginia.edu LIRMM, UNIVMONTPELLIER, CNRS, MONTPELLIER, FRAN...

work page 2025