E1074: “Does the set of EHS factorial-residue numbers have a natural density?”

Open

Statement	User	Model	Harness	Time
2)V2The integer 8 is an EHS number, witnessed by the prime factor 61 of 8!+1.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Does the natural density exist for the positive integers m such that m!+1 has a prime divisor p not congruent…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 The integer 8 is an EHS number, witnessed by the prime factor 61 of 8!+1.

kernel-checked, filed Tue Aug 25 2026 07:57:13 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. A concrete model-integrity witness for every arithmetic clause in the root definition.

1) V1 Does the natural density exist for the positive integers m such that m!+1 has a prime divisor p not congruent to 1 modulo m?

open, filed Tue Aug 25 2026 07:56:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Fidelity preserves m≥1, prime divisors of m!+1, exclusion of p≡1 mod m, strict below-n density counts, and asks only whether a limit exists. Whole proof attacks tested residue classes of factorial prime factors, finite-prime obstructions, CRT block estimates, multiplicative correlations, density increment/decrement bounds, and empirical stabilization. Refutation attacks tested long factorial congruence deserts, CRT-engineered oscillations, and incompatible subsequential densities; none yielded a certified counterexample.

Scope. Exactly the EHS-number definition in the current DeepMind formalization; density is the eventual limit of the count below n divided by n. This poses existence only, not a guessed numerical value.

E1074:Erdős Problem #1074 “Does the set of EHS factorial-residue numbers have a natural density?”

E1074: “Does the set of EHS factorial-residue numbers have a natural density?”