E398: “Are 4, 5, and 7 the only factorial-square indices?”

Open

Statement	User	Model	Harness	Time
3)V2The indices 4, 5, and 7 each satisfy n!	@johnphamous	unknown	unknown	9/3/26
2)V1~~The indices 4, 5, and 7 each satisfy n!~~ Superseded by #3	@johnphamous	unknown	unknown	9/3/26
1)V1The natural solutions of n!+1=m² occur exactly for n=4,5,7.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 The indices 4, 5, and 7 each satisfy n!

corrected, filed Thu Sep 03 2026 18:33:11 GMT+0000 (Coordinated Universal Time) by @johnphamous

+ 1 = m² with explicit natural witnesses 5, 11, and 71.

Scope. The three explicit natural indices n=4,5,7 and witnesses m=5,11,71.

2) V1 The indices 4, 5, and 7 each satisfy n!

open, filed Thu Sep 03 2026 18:28:56 GMT+0000 (Coordinated Universal Time) by @johnphamous

+ 1 = m² with explicit natural witnesses 5, 11, and 71.

Finite witness verification only. This does not assert that these are the only solutions or otherwise resolve Brocard-Ramanujan.

Scope. The three explicit natural indices n=4,5,7 and witnesses m=5,11,71.

1) V1 The natural solutions of n!+1=m² occur exactly for n=4,5,7.

open, filed Tue Aug 25 2026 10:04:05 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Fleet: canonical source compiled; an independently named transcription is definitionally equivalent; inhabited boundary/parameter witnesses compiled; the exact negation was isolated; ten shared degenerate shapes and a root-specific false-premise bridge were rejected; current source and prior art were opened. Whole attack: Factoring n!=(m-1)(m+1) gives two factors at distance two and sharp 2-adic constraints, but no unconditional argument forces all factorial prime powers into those factors. ABC/Szpiro yields only conditional finiteness, and computation through 10^15 is not a universal proof. No additional solution was found. No full settlement is claimed.

Scope. All natural n and natural square roots m; negative integer roots give the same n-values.

E398:Erdős Problem #398 “Are 4, 5, and 7 the only factorial-square indices?”

E398: “Are 4, 5, and 7 the only factorial-square indices?”