REVIEW
The factorial separation conjecture
Number Theory · Submitted 19 Oct 2026 · 14 references
Pith's verdict: Introduces a new conjecture on factorial-perfect power gaps with substantial unconditional results; the fifth-power case is the only incomplete part, but openly acknowledged.
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Referee report
The paper
The paper claims
The Factorial Separation Conjecture proposes that for every integer n ≥ 8, the factorial n! sits at least n away from any perfect power m^k, and proves this inequality for a wide range of exponent and base conditions.
The reading
What carries the argument
The proof combines elementary valuation arguments, Bertrand’s postulate, and Wilson’s theorem with deeper tools: the Erdős–Obláth progression–product estimate (Lemma 3.1), cubic reciprocity in the Eisenstein integers (Lemma 4.2, using the sharp planar Hermite bound from Lemma 4.1), uniform character estimates for fourth powers (Proposition 5.4, relying on Rosser–Schoenfeld bounds and Möbius inversion), and for fifth powers the explicit‑formula technique of Wesolowski (Lemma 6.11) specialized to the Kummer extensions Lₚ = ℚ(ζ₅, $p^{{1/5}}$). The key identity for cubes is the detecting‑prime lemma that uses the geometry of the Eisenstein lattice; for fifth powers the exact conductor pO_K of the Hecke character (Lemma 6.4) is critical for the GRH argument.
Extended reading notes
Core claim
The central discovery is an unconditional proof that the factorials n! for n ≥ 8 are separated from all cubes and all fourth powers (and hence all even powers ≥ 4) by more than n. For cubes the proof uses cubic reciprocity and the planar Hermite bound; for fourth powers it uses a character estimate and a product comparison that avoids any external computation. For fifth powers the paper provides an unconditional finite interval (n ≤ 1100) and an unconditional large‑n argument (with an uncomputed effective threshold N₅), plus a complete proof under a specific GRH hypothesis. These results together show that any counterexample to the full conjecture must be either a square of the form n! = a²(b²−1) or an odd‑prime‑power equation n! + εp = m^ℓ with ℓ ≥ 5 and n/2 < p ≤ n, and also must have P⁺(m) > n+2.
Load-bearing premise
The Generalized Riemann Hypothesis for the Dedekind zeta functions of the fields ℚ(ζ₅, $p^{{1/5}}$) for every prime p ≥ 1100 with p⁴ ≡ 1 mod 25 has not been proved; if this hypothesis is false, the conditional proof of fifth‑power separation for all n ≥ 8 collapses and the full conjecture remains open for fifth powers beyond the known finite interval.
Editorial extensions
If this is right
- If the Factorial Separation Conjecture holds in full, then the Brocard–Ramanujan equation n! + 1 = m² has only the three known solutions (n = 4,5,7).
- For every fixed integer a, the equation n! + a = m^k would have no solutions for n ≥ max{8, |a|}, giving a uniform cutoff independent of the base and exponent.
- The minimal distance from n! to any perfect power would grow at least linearly with n, improving the known super‑exponential lower bound.
- Any putative counterexample must satisfy P⁺(m) > n+2 and must be of the specific forms given in Corollary 7.1: either a square beneath n! (the Brocard–Ramanujan type) or an odd‑prime power with the displacement being a prime between n/2 and n.
- The unconditional results for cubes and even powers already settle the conjecture for all exponents that are not a power of 2 and not coprime to 3 or 5, significantly narrowing the search space.
Reading between the lines
- The reduction of the odd‑prime case to a prime displacement p in (n/2, n] suggests that verifying the conjecture for moderate n could be done by systematically checking finitely many equations per n, possibly with a computational search up to several hundred thousand.
- The unconditional large‑n argument for fifth powers uses an uncomputed threshold N₅; computing N₅ explicitly (even if enormous) would remove the GRH dependence and complete the fifth‑power case unconditionally.
- The technique of using the exact conductor of a Kummer extension to sharpen the explicit‑formula estimate may be extendable to other prime exponents (ℓ = 7, 11, …) to obtain conditional results under GRH for those cases, or to improve unconditional effective thresholds.
- The success of the uniform character estimate for fourth powers suggests that similar product‑comparison arguments might work for other low‑degree cyclotomic fields, potentially covering some of the remaining odd exponents without GRH.
Original abstract
We propose the Factorial Separation Conjecture (FSC), a strict strengthening of earlier factorial gap conjectures: \[ |n!-m^k|>n \] for all integers $n\ge8$ and $m,k\ge2$. The inequality excludes every displacement up to and including the factorial index. We prove the conjecture unconditionally for exponents divisible by three, for even exponents at least four, and for every exponent at least two when the largest prime divisor of the base is at most $n+2$. For exponents divisible by five, we prove the finite interval $8\le n\le1100$ and all sufficiently large indices, with an effective but uncomputed threshold. The conclusion for every $n\ge8$ and every exponent divisible by five is proved under the Generalized Riemann Hypothesis (GRH) for a specified family of Dedekind zeta functions. Here GRH asserts that their nontrivial zeros have real part $1/2$. The arguments combine valuations, cyclotomic factorizations, cubic reciprocity, character estimates and explicit formulas. All finite comparisons are justified in the text. The general conjecture remains open.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We propose the Factorial Separation Conjecture (FSC), a strict strengthening of earlier factorial gap conjectures: \[ |n!-m^k|>n \] for all integers $n\ge8$ and $m,k\ge2$. The inequality excludes every displacement up to and including the factorial index. We prove the conjecture unconditionally for exponents divisible by three, for even exponents at least four, and for every exponent at least two when the largest prime divisor of the base is at most $n+2$. For exponents divisible by five, we prove the finite interval $8\le n\le1100$ and all sufficiently large indices, with an effective but uncomputed threshold. The conclusion for every $n\ge8$ and every exponent divisible by five is proved under the Generalized Riemann Hypothesis (GRH) for a specified family of Dedekind zeta functions. Here GRH asserts that their nontrivial zeros have real part $1/2$. The arguments combine valuations, cyclotomic factorizations, cubic reciprocity, character estimates and explicit formulas. All finite comparisons are justified in the text. The general conjecture remains open.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose the Factorial Separation Conjecture (FSC), a strict strengthening of earlier factorial gap conjectures: \[ |n!-m^k|>n \] for all integers $n\ge8$ and $m,k\ge2$. | contribution 0.98 | 0.78 | 0.38 | speculative 0.85 |
| s1 | The inequality excludes every displacement up to and including the factorial index. | interpretation 0.91 | 0.29 | 0.08 | asserted 0.99 |
| s2 | We prove the conjecture unconditionally for exponents divisible by three, for even exponents at least four, and for every exponent at least two when the largest prime divisor of the base is at most $n+2$. | result 0.93 | 0.94 | 0.24 | asserted 1.00 |
| s3 | For exponents divisible by five, we prove the finite interval $8\le n\le1100$ and all sufficiently large indices, with an effective but uncomputed threshold. | result 0.93 | 0.94 | 0.08 | asserted 1.00 |
| s4 | The conclusion for every $n\ge8$ and every exponent divisible by five is proved under the Generalized Riemann Hypothesis (GRH) for a specified family of Dedekind zeta functions. | result 0.87 | 0.84 | 0.20 | asserted 1.00 |
| s5 | Here GRH asserts that their nontrivial zeros have real part $1/2$. | background 0.95 | 0.11 | 0.02 | asserted 0.98 |
| s6 | The arguments combine valuations, cyclotomic factorizations, cubic reciprocity, character estimates and explicit formulas. | method 1.00 | 0.81 | 0.03 | asserted 1.00 |
| s7 | All finite comparisons are justified in the text. | method 0.40 | 0.76 | 0.03 | asserted 1.00 |
| s8 | The general conjecture remains open. | limitation 0.94 | 0.35 | 0.23 | asserted 0.98 |
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References
Discussion
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