CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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$Γ (n + 1) = n!$ for every natural number $n$

Statement

For every natural number $n$ , $Γ (n + 1) = n!$ .

Facts & Assumptions

Given: Factorial recursion $0! = 1$ and $(n + 1)! = (n + 1) n!$ from The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ .

[F1]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ , and $Γ (1) = 1$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

Proof

technique · induction

1.1

At $n = 0$ , [F1] gives $Γ (1) = 1 = 0!$ .

F1base

1.2

Assume $Γ (n + 1) = n!$ . Then [F1] gives $Γ (n + 2) = (n + 1) Γ (n + 1) = (n + 1) n! = (n + 1)!$ .

F1ihalgebra

2.1

The induction principle therefore proves $Γ (n + 1) = n!$ for every $n \in N$ .

step 1.1step 1.2discharge-induction∎

Depends on

Used by

The real Gamma function has one minimum and diverges at both ends of its domain Theorem

Dependency tree · two levels

24 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1.2 (standard reference, not scraped)

Γ(n+1)=n!Γ(n+1)=n! for every natural number nn