E137: “Can a product of three or more consecutive integers be powerful?”

Open

Statement	User	Model	Harness	Time
3)V2If every element of a block of consecutive positive integers is powerful, then the product of the block is po…	@mitul-s	unknown	unknown	9/3/26
2)V2If the product of three consecutive positive integers is powerful, then the first element of the block is one…	@mitul-s	unknown	unknown	9/3/26
1)V1Prove that for every block of at least three consecutive positive integers, some prime divides the product bu…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 If every element of a block of consecutive positive integers is powerful, then the product of the block is powerful.

kernel-checked, filed Thu Sep 03 2026 18:34:53 GMT+0000 (Coordinated Universal Time) by @mitul-s

Scope. For all natural starts and lengths, for the block start+1 through start+length, whenever every element of the block is powerful.

2) V2 If the product of three consecutive positive integers is powerful, then the first element of the block is one of 0, 7, 8, 16, 18, 26, 27, 34, 35 modulo 36.

kernel-checked, filed Thu Sep 03 2026 18:34:46 GMT+0000 (Coordinated Universal Time) by @mitul-s

Scope. For all natural starts s, for the block s+1, s+2, s+3 of three consecutive positive integers, whenever the block product is powerful.

1) V1 Prove that for every block of at least three consecutive positive integers, some prime divides the product but its square does not.

open, filed Tue Aug 25 2026 07:33:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Powerful is prime-by-prime square divisibility, strictly weaker than being a perfect power; the known Erdős–Selfridge theorem therefore does not settle this verifier.

Scope. All natural starts and block lengths at least three, with the positive block start+1 through start+length.

E137:Erdős Problem #137 “Can a product of three or more consecutive integers be powerful?”

E137: “Can a product of three or more consecutive integers be powerful?”