[#P2844] Classify factorials in the interior of Pascal's triangle

Problem. Determine every triple of integers \((n,k,m)\) with \(n\ge4\), \(2\le k\le n/2\), and \(m\ge2\) such that \(\binom nk=m!\). Prove that the resulting list is complete.

Agent access

Work on this problem in ChatGPT

Definitions and notation

1Context

This elementary-looking classification mixes prime divisors of consecutive products with factorial valuations. Search tables of smooth binomial coefficients and congruence obstructions can be reused as the unconditional cutoff improves.

2Problem setup

Definition 1 (The binomial coefficient). The binomial coefficient is \(\binom nk=n!/(k!(n-k)!)\).

Definition 2 (The condition \(2\le k\le n/2\) chooses one representative under \(k\leftrightarrow n-k\) and excludes the two boundary diagonals). The condition \(2\le k\le n/2\) chooses one representative under \(k\leftrightarrow n-k\) and excludes the two boundary diagonals.

Definition 3 (Completeness requires a proof covering all unbounded values of \(n\), \(k\), and \(m\), rather than a finite search alone). Completeness requires a proof covering all unbounded values of \(n\), \(k\), and \(m\), rather than a finite search alone.

Remark 1. This elementary-looking classification mixes prime divisors of consecutive products with factorial valuations. Search tables of smooth binomial coefficients and congruence obstructions can be reused as the unconditional cutoff improves.

3What counts as a solution

List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.
If a finite computation is part of the proof, provide an exact certificate for the searched range and a theorem that reduces every remaining case to that finite range. Conditional abc arguments do not meet acceptance.

1Status

What counts as a solution

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis. List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis. List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis. List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.^[1] Relevance to this problem For factorials interior pascal triangle, pins the dated research frontier: UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis. List all triples. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis. A complete resolution must satisfy this condition: List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist. Open full record →
List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist.^[1] Relevance to this problem For Classify factorials in the interior of Pascal's triangle, record factorials-interior-pascal-triangle-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. List all triples satisfying the stated inequalities, verify each identity, and prove that no further triples exist. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. UNKNOWN as of 2026-07-31. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis.

On 2026-07-27 all five MathOverflow answers and their comments were checked. Computations and necessary smoothness tests report no additional solutions in their stated ranges, while none supplies an unconditional tail argument.
The three reported representatives under \(k\leftrightarrow n-k\) are \((4,2,3)\), \((10,3,5)\), and \((16,2,5)\). Any claimed classification must verify these directly and cover every larger parameter.
Nair and Shorey's paper, DOI 10.1016/j.indag.2015.12.002, treats related factorial-product equations under a strong explicit abc conjecture. Its conditional conclusion cannot certify the unconditional classification here.
For \(\binom nk=m!\), every prime divisor of the binomial coefficient is at most \(m\). Smoothness filters can rule out candidates, but passing the filter does not establish the factorial identity.
Trap: the edges \(k=1,n-1\) contain every factorial through \(\binom{m!}{1}=m!\). They are excluded by the interior condition.

Recorded example 1. \(\binom42=6=3!\), \(\binom{10}3=120=5!\), and \(\binom{16}2=120=5!\).

Recorded example 2. The equality \(\binom{m!}{1}=m!\) lies on a boundary diagonal and is outside the stated range.

Computational notes

The MathOverflow thread reports direct and smoothness-based searches in finite ranges. Those reports are evidence only because they do not supply an unconditional reduction of the infinite target.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemClassify factorials in the interior of Pascal's triangle

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Classify factorials in the interior of Pascal's triangle.” TheoremDB. P2844. Problem statement; statement text SHA-256 3dd6c4ca13c7eff3a3ab261d0ff17b1eb8f0d79769ce71dfa1dfbdc09ad2c22a. https://theoremdb.org/statement/?ref=P2844

BibTeX

@misc{theoremdb-problem-3dd6c4ca13c7eff3a3ab261d0ff17b1eb8f0d79769ce71dfa1dfbdc09ad2c22a,
  title = {{Classify factorials in the interior of Pascal's triangle}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 3dd6c4ca13c7eff3a3ab261d0ff17b1eb8f0d79769ce71dfa1dfbdc09ad2c22a},
  url = {https://theoremdb.org/statement/?ref=P2844}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Factorials in Pascal's triangle, MathOverflow question 17058. Original CC0 classification statement written after reading the question, all five answers, comments, and the later conditional finiteness literature cited in the thread. mathoverflow.net checked 2026-08-01. Original CC0 classification statement written after reading the question, all five answers, comments, and the later conditional finiteness literature cited in the thread. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Classify factorials in the interior of Pascal's triangle: UNKNOWN as of 2026-07-27. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis.Source named by the research packet.
Saranya G. Nair and T.N. Shorey, “On the equation n ! = a 1 ! a 2 ! ⋯ a t !”. Indagationes Mathematicae 27(3) (2016), 634-642. DOI 10.1016/j.indag.2015.12.002. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.UNKNOWN as of 2026-07-27. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis.Also cited at Full journal article relevant to Classify factorials in the interior of Pascal's triangle.Source used to assess the problem's recorded status.For Classify factorials in the interior of Pascal's triangle: UNKNOWN as of 2026-07-27. The checked thread records the interior solutions \(\binom42=3!\), \(\binom{10}3=5!\), and \(\binom{16}2=5!\), but no unconditional completeness proof. Nair and Shorey obtain conditional finiteness results assuming a strong explicit abc-type hypothesis.

Original self-contained restatement motivated by MathOverflow question 17058; it fixes symmetry and boundary conventions.

[#P2844] Classify factorials in the interior of Pascal's triangle

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance