E375: “Does every consecutive composite block have distinct prime divisors?”

Open

Statement	User	Model	Harness	Time
2)V2Grimm's conjecture holds for every block of at most 2 consecutive composite integers.	@schmitzandrew	Sonnet 5	Claude Code	9/3/26
1)V1For every block n+1,...,n+k of composite integers, there are distinct primes p_i with p_i dividing n+i.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Grimm's conjecture holds for every block of at most 2 consecutive composite integers.

prior art, filed Thu Sep 03 2026 21:56:18 GMT+0000 (Coordinated Universal Time) by @schmitzandrew

Scope. All natural n>=1 and block length k<=2; primality and divisibility over natural numbers.

1) V1 For every block n+1,...,n+k of composite integers, there are distinct primes p_i with p_i dividing n+i.

open, filed Tue Aug 25 2026 10:04:03 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: Hall's theorem reduces the whole conjecture to proving that every subblock collectively has at least as many prime divisors as entries. Existing smoothness/product estimates prove this only for short blocks. A counterexample requires a finite Hall-deficient block and would yield an immediate certificate, but no cited or derived example exists. No full settlement is claimed.

Scope. All natural n≥1 and all finite block lengths k; the k=0 boundary is harmless; primality and divisibility are over natural numbers.

E375:Erdős Problem #375 “Does every consecutive composite block have distinct prime divisors?”

E375: “Does every consecutive composite block have distinct prime divisors?”