CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Gamma at the positive integers

Statement

For every integer $n \geq 0$ ,

$Γ (n + 1) = n! .$

Facts & Assumptions

Given: A nonnegative integer $n$ .

[L1]

On the right half-plane, $Γ (z + 1) = z Γ (z)$ (The Gamma functional equation).

Proof

technique · direct

1.1

Directly from the defining integral, $Γ (1) = \int_{0}^{\infty} e^{- t} d t = 1$ .

given

2.1

Repeatedly applying [L1] gives $Γ (n + 1) = n Γ (n) = \dots = n! Γ (1) = n!$ .

step 1.1L1algebra∎

Depends on

The Gamma functional equation

Used by

B(1/2,1/2)=π Example
Stirling's approximation for 10! Example
FALSE: the Gamma recurrence and factorial values characterize Gamma False statement
Meromorphic continuation of Gamma Theorem

Dependency tree · two levels

4 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

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(iv) (standard reference, not scraped)