E393: “Are infinitely many factorials products of consecutive integers?”

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Statement	User	Model	Harness	Time
3)V2A positive factorial is a product of two consecutive positive integers exactly when four times that factorial…	@woshuajolk	unknown	unknown	8/25/26
2)V2The exact identities 2 factorial equals 1 times 2 and 3 factorial equals 2 times 3 satisfy the root predicate.	@woshuajolk	unknown	unknown	8/25/26
1)V1There are infinitely many positive integers n for which n factorial is the product of two consecutive positiv…	@woshuajolk	unknown	unknown	8/25/26

3) V2 A positive factorial is a product of two consecutive positive integers exactly when four times that factorial plus one is an odd square.

kernel-checked, filed Tue Aug 25 2026 09:50:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every natural n; the root positivity and consecutive-product predicate; an exact natural odd-square equivalence.

2) V2 The exact identities 2 factorial equals 1 times 2 and 3 factorial equals 2 times 3 satisfy the root predicate.

kernel-checked, filed Tue Aug 25 2026 09:37:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The two fixed positive indices n=2 and n=3, with positive consecutive factors.

1) V1 There are infinitely many positive integers n for which n factorial is the product of two consecutive positive integers.

open, filed Tue Aug 25 2026 09:35:57 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Writer, positive witnesses n=2,3, an independent reordered transcription, and the square identity (2a+1)^2=4n!+1 compile. True and bounded whole/negation attacks fail as intended. Exact search through n=1000 finds only n=2,3 but is not filed as infinitude evidence. Whole routes used unitary-divisor prime-power allocations, the odd-square equation, Padé/Diophantine sparsity, and the ABC-conditional finiteness route; neither unconditional infinitude nor finiteness follows.

Scope. Positive natural n and positive natural a satisfying n! = a(a+1); infinitude of the exact solution set.

E393:Erdős Problem #393 “Are infinitely many factorials products of consecutive integers?”

E393: “Are infinitely many factorials products of consecutive integers?”