False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: the quotient $(\binom{2 n}{n}) / (n + 1)$ is an integer only for small $n$

Statement

False claim: the quotient

$\frac{(\binom{2 n}{n})}{n + 1}$

is an integer only for small values of $n$ .

Facts & Assumptions

Given: a natural number $n$ .

[L1]

$(n + 1) C_{n} = (\binom{2 n}{n})$ ( $(n + 1) C_{n} = (\binom{2 n}{n})$ ).

[L2]

$n + 1$ divides $(\binom{2 n}{n})$ for every $n$ ( $n + 1$ divides $(\binom{2 n}{n})$ for every $n \in N$ ).

Refutation

technique · direct

1.1

The first values of the quotient are $1, 1, 2, 5, 14, 42, 132$ at $n = 0, 1, 2, 3, 4, 5, 6$ respectively, so the quotient keeps producing integers beyond the first few cases.

2.1

More generally, [L1] rewrites the quotient as $C_{n}$ for every natural number $n$ , and $C_{n}$ is a natural number by definition. So the quotient is an integer for every $n$ , not merely for small ones.

L1L2∎

Remarks

The point of the refutation is that the divisibility is proved by exhibiting a count. Once the quotient is $C_{n}$ , no separate arithmetic argument is needed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Guichard, An Introduction to Combinatorics and Graph Theory, §3.5 (standard reference, not scraped)

FALSE: the quotient (2nn)/(n+1)(n2n​)/(n+1) is an integer only for small nn