Pith. sign in

REVIEW 2 major objections 1 minor 68 references

A comment on the equation $n!!=a_1!!\cdots a_t!!$

T0 review · 2 major / 1 minor · reviewed 2026-05-10 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The explicit abc conjecture implies only finitely many nontrivial solutions to the double factorial equation in certain special cases.

desk verdict A short note deriving conditional finiteness for the double factorial equation from the abc conjecture in a few special cases. read the letter →

arxiv 2604.09730 v1 submitted 2026-04-09 math.NT

Saša Novaković This is my paper · ORCID
classification math.NT
keywords doublefactorialabcconjectureDiophantineequationfinitenessofsolutionsnumbertheory
topics The abc Conjecture
open problems The abc Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper examines the equation expressing one double factorial as a product of several others. It focuses on special cases and applies the explicit form of the abc conjecture to prove that only finitely many nontrivial solutions exist. A reader would care because this links a concrete combinatorial identity to a major arithmetic conjecture about the distribution of primes and radicals. If the implication holds, the equation cannot have arbitrarily large or unexpected solutions beyond a finite list.

What carries the argument

The explicit abc conjecture applied to the radicals and prime factors that arise when double factorials are expanded in the equation.

What would settle it

A concrete list of infinitely many distinct nontrivial solutions in one of the special cases covered by the claim would show the implication fails.

Watch

Extended reading notes

Core claim

In certain special cases the explicit abc conjecture implies that the equation a1!!⋯at!!=n!! has only finitely many nontrivial solutions.

Load-bearing premise

That the explicit abc conjecture applies directly to the unspecified special cases of the double factorial equation and that the resulting implication holds without additional unstated conditions.

Editorial extensions

If this is right

  • The set of solutions is finite once the special cases are fixed.
  • Any solution must satisfy a bound derived from the abc quality measure.
  • Trivial decompositions such as single-term products are excluded from the finiteness statement.

Reading between the lines

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

  • The same reduction might extend to equations involving multifactorials or ordinary factorials under analogous conjectures.
  • Small explicit solutions could be enumerated by computer to complete the classification if abc is assumed.
  • The approach treats the double factorial as a source of prime factors whose product of (p-1) terms controls the radical.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the Diophantine equation a1!! ⋯ at!! = n!! and asserts that, under the explicit abc conjecture, the equation has only finitely many nontrivial solutions in certain special cases.

Significance. If the special cases were made explicit and a uniform reduction to an abc-suitable a+b=c form were established, the result would supply a conditional finiteness statement for a variant of the factorial equation. The current text supplies neither the cases nor the reduction, so the potential contribution cannot yet be evaluated.

major comments (2)
  1. [Abstract] Abstract: the phrase 'certain special cases' is never defined or delimited (e.g., by fixing t, imposing parity conditions on the ai, or restricting n to an arithmetic progression). This definition is load-bearing for the central claim, because the radical bound required by the explicit abc conjecture may fail to be uniform once the double-factorial denominators are expanded.
  2. [Main text] Main text: no derivation is supplied that rewrites the product of double factorials into an a+b=c equation in which rad(abc) ≪ max(|a|,|b|,|c|)^ε for some ε<1 independent of n. Without this step the implication from the explicit abc conjecture cannot be verified.
minor comments (1)
  1. [Abstract] The term 'nontrivial solutions' is used but never defined; a brief clarification would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which identify important gaps in clarity and detail. We will revise the manuscript to explicitly define the special cases and supply the required derivation, thereby strengthening the presentation of the conditional finiteness result.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the phrase 'certain special cases' is never defined or delimited (e.g., by fixing t, imposing parity conditions on the ai, or restricting n to an arithmetic progression). This definition is load-bearing for the central claim, because the radical bound required by the explicit abc conjecture may fail to be uniform once the double-factorial denominators are expanded.

    Authors: We agree that the special cases must be made explicit. In the revised version we will delimit them by fixing t, imposing parity conditions on the a_i (for instance requiring all a_i even), and restricting n to suitable arithmetic progressions. These restrictions ensure that the double-factorial expansions produce a uniform radical bound compatible with the explicit abc conjecture. revision: yes

  2. Referee: [Main text] Main text: no derivation is supplied that rewrites the product of double factorials into an a+b=c equation in which rad(abc) ≪ max(|a|,|b|,|c|)^ε for some ε<1 independent of n. Without this step the implication from the explicit abc conjecture cannot be verified.

    Authors: We acknowledge that the current text omits the explicit reduction. The revised manuscript will contain a dedicated derivation showing how the equation a_1!! ⋯ a_t!! = n!! can be rewritten, under the chosen special cases, as an a + b = c instance satisfying rad(abc) ≪ max(|a|,|b|,|c|)^ε with ε < 1 independent of n. This step will make the application of the explicit abc conjecture fully verifiable. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: implication from external abc conjecture to finiteness in special cases

full rationale

The paper's central claim is an implication: in unspecified special cases of the double-factorial product equation, the explicit abc conjecture yields only finitely many nontrivial solutions. This is not self-definitional, does not rename a fitted quantity as a prediction, and does not rest on a load-bearing self-citation chain. The abc conjecture is an independent external hypothesis; any rewriting of a1!!⋯at!!=n!! into a+b=c form with controlled radical would be a separate (possibly missing) derivation step, but does not reduce the stated result to the paper's own inputs by construction. The derivation chain is therefore self-contained against the external benchmark.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the explicit abc conjecture as an external assumption. No free parameters or invented entities are mentioned in the abstract.

assumptions (1)
  • domain assumption The explicit abc conjecture
    Invoked to derive the finiteness of nontrivial solutions in the special cases.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A comment on the equation $n!!=a_1!!\cdots a_t!!$." pith.science (2026). https://pith.science/paper/2604.09730

@misc{pith2026260409730,
  author       = {Pith},
  title        = {Pith review of: A comment on the equation $n!!=a_1!!\cdots a_t!!$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.09730}},
  note         = {Machine review of arXiv:2604.09730}
}
abstract

We study the equation $a_1!!\cdots a_t!!=n!!$ and show that in certain special cases the explicit abc conjecture implies that it has only finitely many nontrivial solutions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    Alzer and F

    H. Alzer and F. Luca, Diophantine equations involving factorials. Mathematica Bohemica 4 (2017), p. 181-184

    work page 2017

  2. [2]

    Baker: A sharpening of the bounds for linear forms in logarithms

    A. Baker: A sharpening of the bounds for linear forms in logarithms. Acta Arith. 21 (1972), 117-129

    work page 1972

  3. [3]

    Bachman, Introduction of p -adic numbers and valuation theory, Academic Press, New York, 1964

    G. Bachman, Introduction of p -adic numbers and valuation theory, Academic Press, New York, 1964

    work page 1964

  4. [4]

    Baczkowski, M

    D. Baczkowski, M. Filaseta, F. Luca and O. Trifonov, On values of d(n!)/m! , (n!)/m! and (n!)/m! . Int. J. Number Theory 6 (2010), 1199-1214

    work page 2010

  5. [5]

    Baczkowski and S

    D. Baczkowski and S. Novakovi\'c, Some diophantine equations involving arithmetic functions and Bhargava factorials. Colloqium Mathematicum 177 (2024), 21-30

    work page 2024

  6. [6]

    Baker, Experiments on the abc-conjecture

    A. Baker, Experiments on the abc-conjecture. Publ. Math. Debrecen 65 (2004), 253-260

    work page 2004

  7. [7]

    R. C. Baker, G. Harman\ and\ J. Pintz, The difference between consecutive primes. II , Proc. London Math. Soc. (3) 83 (2001), no. 3, 532--562

    work page 2001

  8. [8]

    Bennett, et al., Explicit bounds for primes in arithmetic progressions

    M. Bennett, et al., Explicit bounds for primes in arithmetic progressions. Illinois J. Math. (2018)

    work page 2018

Show all 68 references
  1. [9]

    R. C. Baker, G. Harman, J. Pintz: The Difference Between Consecutive Primes, II, Proceedings of the London Mathematical Society, Volume 83, Issue 3, (2001), 532--562

    2001

  2. [10]

    Bhargava, P-orderings and polynimial functions on arbitrary subsets of Dedekind rings

    M. Bhargava, P-orderings and polynimial functions on arbitrary subsets of Dedekind rings. J. Reine Angew. Math. 490 (1997), 101-127

    1997

  3. [11]

    Berend and C.F

    D. Berend and C.F. Osgood, On the equation P(x)=n! and a question of Erd o s. J. Number Theory. 42 (1992), 189-193

    1992

  4. [12]

    Berend and J.E

    D. Berend and J.E. Harmse, On polynomial-factorial Diophantine equations. Trans. Amer. Math. Soc. 358 (2006), 1741-1779

    2006

  5. [13]

    Berndt and W.F

    B.C. Berndt and W.F. Galway, On the Brocard-Ramanujan Diophantine equation n!+1=m^2 . The Ramanujan J. 4 (2000), 41-42

    2000

  6. [14]

    Brocard: Question 1532

    H. Brocard: Question 1532. Nouv. Corresp. Math. 2 (1876); Nouv. Ann. Math. 4 (1885), 391

  7. [15]

    Dusart, Autour de la fonction qui compte le nombre de nombre premiers

    P. Dusart, Autour de la fonction qui compte le nombre de nombre premiers. Ph.D thesis, Universit\'e de Limoges (1998)

    1998

  8. [16]

    Dusart, In\'egalit\'es explicites pour (X), (X), (X) et les nombres premiers

    P. Dusart, In\'egalit\'es explicites pour (X), (X), (X) et les nombres premiers. C.R. Math. Acad. Sci. Soc. R. Can. 21 (1999), 53-59

    1999

  9. [17]

    Erd o s and R

    P. Erd o s and R. Obl\'ath, \"Uber diophantische Gleichungen der Form n!=x^p y^p und n! m!= x^p . Acta Szeged. 8 (1937), 241-255

    1937

  10. [18]

    Erd o s, On consecutive integers

    P. Erd o s, On consecutive integers. Nieuw. Arch. Wiskd. 3 (1955), 124-128

    1955

  11. [19]

    Erd o s\ et al., On the number of divisors of n! , in Analytic number theory, Vol

    P. Erd o s\ et al., On the number of divisors of n! , in Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) , 337--355, Progr. Math., 138, Birkh\" a user Boston, Boston, MA

    1995

  12. [20]

    Bugeaud, M

    Y. Bugeaud, M. Mignotte and S. Siksek: Classical and modular aproaches to exponential and Diophantine equations II. The Lebesgue--Nagel equation. Compos. Math. 142 (2006), 31-62

    2006

  13. [21]

    H.M. Bui, K. Pratt\ and\ A. Zaharescu, Power savings for counting solutions to polynomial-factorial equations, Adv. Math. 422 (2023), Paper No. 109021, 32 pp

    2023

  14. [22]

    Dabrowski, On the equation n!+A=y^2

    A. Dabrowski, On the equation n!+A=y^2 . Nieuw Arch. Wisk. 14 (1996), 321-324

    1996

  15. [23]

    Dabrowski: On the Brocard--Ramanujan problem and generalizations

    A. Dabrowski: On the Brocard--Ramanujan problem and generalizations. Coll. Mathe. 126 (2012), 105-110

    2012

  16. [24]

    Dabrowski and M

    A. Dabrowski and M. Ulas, Variations on the Brocard--Ramanujan equation. J. Number Theory 133 (2013), 1168-1185

    2013

  17. [25]

    Epstein and J

    A. Epstein and J. Glickman (2020), https://github.com/jhg023/brocard

    2020

  18. [26]

    Dufour and O

    B. Dufour and O. Kihel, The Brocard--Ramanujan equation. Int. J. Math 5 (2004), 577-580

    2004

  19. [27]

    K. Ford, F. Luca and C. Pomerance, Common values of the arithmetic function and . Bull. Lond. Math. Soc. 42 (2010), 478-488

    2010

  20. [28]

    Hardy and E.M

    G.H. Hardy and E.M. Wright, An introduction to the theory of number (5th ed.), Oxford Univ. Press, New York (1979)

    1979

  21. [29]

    Janusz, Algebraic number fields, Academic Press, New York and London (1973)

    G.J. Janusz, Algebraic number fields, Academic Press, New York and London (1973)

    1973

  22. [30]

    Erd o s and R.L

    P. Erd o s and R.L. Graham, Old and new problems and resolutions in combinatorial number theory, Monography No. 28 L'Enseignement Math. Geneve (1980)

    1980

  23. [31]

    Kihel and F

    O. Kihel and F. Luca, Variants of the Brocard--Ramanujan equation. J. Th\'eor. Nombres Bordeaux 20 (2008), 353-363

    2008

  24. [32]

    Lang: Old and new conjectured diophantine inequalities

    S. Lang: Old and new conjectured diophantine inequalities. Bull. Amer. Math. Soc. 23 (1990), 37-75

    1990

  25. [33]

    Halberstam and H.-E

    H. Halberstam and H.-E. Richert, Sieve Methods , London Mathematical Society Monographs, No. 4. Academic Press, 1974

    1974

  26. [34]

    Gawron, A note on the Diophantine equation P(z)=n!+m!

    M. Gawron, A note on the Diophantine equation P(z)=n!+m! . Coll. Mathe. 131 (2013), 53-58

    2013

  27. [35]

    Guy, Unsolved problems in number theory

    R.K. Guy, Unsolved problems in number theory. Springer, New York (1994)

    1994

  28. [36]

    Lang, Old and new conjectured Diophantine inequalities

    S. Lang, Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc. 23 (1990), 37-75

    1990

  29. [37]

    Luca, Equations involving arithmetic functions of factorials

    F. Luca, Equations involving arithmetic functions of factorials. Divulgaciones Math. 8 (2000), 15-23

    2000

  30. [38]

    Laishram and T.N

    S. Laishram and T.N. Shorey, Baker's explicit abc-conjecture and applications. Acta Arith. 155 (2012), 419-429

    2012

  31. [39]

    Luca, The Diophantine equation P(x)=n! and a result of M

    F. Luca, The Diophantine equation P(x)=n! and a result of M. Overholt. Glasnik Matemati\'cki 37 (2002), 269-273

    2002

  32. [40]

    Luca, On factorials which are products of factorials

    F. Luca, On factorials which are products of factorials. Math. Proc. Camb. Phil. Soc. (2007), 143-533

    2007

  33. [41]

    F. Luca, N. Saradha and T.N. Shorey, Squares and factorials in products of factorials. Monatsh. Math. (2014), 385-400

    2014

  34. [42]

    Luca, On the Diophantine equations f(n)=u!+v!

    F. Luca, On the Diophantine equations f(n)=u!+v! . Glasnik Matemati\'cki 48 (2013), 31-48

    2013

  35. [43]

    Matson, Brocard's problem 4th solution search utilizing quadratic residues

    R. Matson, Brocard's problem 4th solution search utilizing quadratic residues. Unsolved Problems in Number Theory, Logic and Cryptography (2017), available at http://unsolvedproblems.org/S99.pdf

    2017

  36. [44]

    Makki Naciri, On the variant Q(n!)=P(x) of the Brocard--Ramanujan Diophantine equation

    A. Makki Naciri, On the variant Q(n!)=P(x) of the Brocard--Ramanujan Diophantine equation. Ramanujan J. 65 (2024), 1791-1798

    2024

  37. [45]

    Makki Naciri, On the Brocard-Ramanujan equation with 7-free integers and prime powers

    A. Makki Naciri, On the Brocard-Ramanujan equation with 7-free integers and prime powers. Integers 25 (2025)

    2025

  38. [46]

    S. G. Nair and T.N. Shorey, Lower bounds for the greatest prime factor product of consecutive positive integers. J. Number Theory 159 (2016), 307-328

    2016

  39. [47]

    Nair and T.N Shorey, On products from blocks of consecutive odd primes

    S.G. Nair and T.N Shorey, On products from blocks of consecutive odd primes. Publ. Math. Debrecen 92 (2018), 1-15

    2018

  40. [48]

    Novakovi\'c, Diophantine equations involving double factorials, arXiv:2510.24312

    S. Novakovi\'c, Diophantine equations involving double factorials, arXiv:2510.24312

  41. [49]

    Novakovi\'c, A note on some polynomial-factorial Diophantine equations

    S. Novakovi\'c, A note on some polynomial-factorial Diophantine equations. Glasnik Matematicki 60 (2025), 21-38

    2025

  42. [50]

    Novakovi\'c, On a generalization of the Brocard--Ramanujan Diophantine equation, arXiv:2602.09678

    S. Novakovi\'c, On a generalization of the Brocard--Ramanujan Diophantine equation, arXiv:2602.09678

  43. [51]

    Novakovi\'c, On the Diophantine equation An!+Bm!=f(x,y)

    S. Novakovi\'c, On the Diophantine equation An!+Bm!=f(x,y) . arXiv:2309.15007

  44. [52]

    Novakovi\'c, The Diophantine equation P(x)= r i=1 H_ n_i , arXiv:2601.16757

    S. Novakovi\'c, The Diophantine equation P(x)= r i=1 H_ n_i , arXiv:2601.16757

  45. [53]

    Overholt, The Diophantine equation n!+1=m^2

    M. Overholt, The Diophantine equation n!+1=m^2 . Bull. London. Math. Soc. 42 (1993), 104

    1993

  46. [54]

    Pollack and H.N

    R.M. Pollack and H.N. Shapiro, The next to last case of a factorial Diophantine equation. Comm. Pure Appl. Math. 25 (1973), 313-325

    1973

  47. [55]

    Ramanujan, Question 469

    S. Ramanujan, Question 469. J. Indian Math. Soc. 5 (1913), 59

    1913

  48. [56]

    Rosser and L

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

    1962

  49. [57]

    Saunders, Diophantine equations involving the Euler totient function

    J.C. Saunders, Diophantine equations involving the Euler totient function. J. Number Theory 209 (2020), 347-358

    2020

  50. [58]

    Shorey: Diophantine approximations, Diophantine equations, transcendence and applications

    T.N. Shorey: Diophantine approximations, Diophantine equations, transcendence and applications. Indian Jour. of Pure and Applied Math. 37 (2006), 9-39

    2006

  51. [59]

    Shorey and R

    T.N. Shorey and R. Tijdeman: Exponential Diophantine equations, Cambridge Tracts in Math. 87, Cambridge University Press (1986)

    1986

  52. [60]

    Siksek: Diophantine equations after Fermat's last theorem

    S. Siksek: Diophantine equations after Fermat's last theorem. J. Th\'eor. Nombres Bordeaux 21 (2009), 425-436

    2009

  53. [61]

    Takeda, Finiteness of trivial solutions of factorial products yielding a factorial over number fields

    W. Takeda, Finiteness of trivial solutions of factorial products yielding a factorial over number fields. Acta Arith. 190 (2019), 395-401

    2019

  54. [62]

    Takeda, On the finiteness of solutions for polynomial-factorial Diophantine equations

    W. Takeda, On the finiteness of solutions for polynomial-factorial Diophantine equations. Forum Math. 33 (2021), 361-374

    2021

  55. [63]

    Takeda, Product of Factorials Equal Another Product of Factorials

    W. Takeda, Product of Factorials Equal Another Product of Factorials. Bull. Iranian Math. Soc. 50 (2024), 1-23

    2024

  56. [64]

    Siegel: Aproximation algebraischer Zahlen

    C.L. Siegel: Aproximation algebraischer Zahlen. Math. Zeit. 10 (1921), 173-213

    1921

  57. [65]

    Tijdeman: Applications of the Gel'fond--Baker method to rational number theory

    R. Tijdeman: Applications of the Gel'fond--Baker method to rational number theory. Colloquia Math. Soc. J\'anos Bolyai (13), Topics in number theory (1974), 339-416

    1974

  58. [66]

    Ulas, Some observations on the Diophantine equation y^2=x!+A and related results

    M. Ulas, Some observations on the Diophantine equation y^2=x!+A and related results. Bull. Aust. Math. Soc. 86 (2012), 377-388

    2012

  59. [67]

    Ulas, Some experiments with Ramanujan--Nagell type Diophantine equations

    M. Ulas, Some experiments with Ramanujan--Nagell type Diophantine equations. Glasnik Matemati\'cki 49 (2014), 287-302

    2014

  60. [68]

    Yamada, A generalization of the Ramanujan--Nagell equation

    T. Yamada, A generalization of the Ramanujan--Nagell equation. Glasgow Math. J. 61 (2019), 535-544

    2019

Pith tools

Reviewed May 10, 2026 · model on record in the stance chip above.