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arxiv: 2604.09730 · v1 · submitted 2026-04-09 · 🧮 math.NT

A comment on the equation n!!=a₁!!cdots a_t!!

Sa\v{s}a Novakovi\'c This is my paper

Pith reviewed 2026-05-10 18:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords double factorialabc conjectureDiophantine equationfiniteness of solutionsnumber theory
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The pith

The explicit abc conjecture implies only finitely many nontrivial solutions to the double factorial equation in certain special cases.

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

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.

Core claim

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

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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.

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.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

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

pith-pipeline@v0.9.0 · 5314 in / 1228 out tokens · 73909 ms · 2026-05-10T18:12:35.516397+00:00 · methodology

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