E25: “Must every truncated congruence sieve have logarithmic density?”

Open

Statement	User	Model	Harness	Time
3)V2Below the kth modulus, the infinite thresholded survivor predicate is equivalent to checking only congruences…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The empty set of natural numbers has logarithmic density zero under the normalization used in the root statem…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every strictly increasing sequence of positive moduli and arbitrary associated residue classes, the integ…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 Below the kth modulus, the infinite thresholded survivor predicate is equivalent to checking only congruences with indices less than k.

kernel-checked, filed Tue Aug 25 2026 03:46:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every strictly increasing modulus sequence seq_n, every residue sequence seq_a, and all k,x ∈ ℕ satisfying x < seq_n k.

2) V2 The empty set of natural numbers has logarithmic density zero under the normalization used in the root statement.

kernel-checked, filed Tue Aug 25 2026 03:44:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The single set ∅ ⊆ ℕ and density value 0, using exactly the root logarithmic-density formula.

1) V1 For every strictly increasing sequence of positive moduli and arbitrary associated residue classes, the integers surviving each congruence only after its modulus threshold have a logarithmic density.

open, filed Tue Aug 25 2026 03:44:15 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Writer mapped every quantifier to the current formal-conjectures RHS. Degenerate True artifact built but reded locally at anti-restatement. No negation was found. Witness seq_n(i)=i+1 is positive and strictly monotone, and x=0 satisfies the survivor predicate. Independent transcription is definitionally equivalent in both directions. All listed sources were opened; vendor diversity was unavailable.

Scope. All strictly increasing sequences seq_n : ℕ → ℕ of positive moduli and all residue sequences seq_a : ℕ → ℤ; the survivor set uses the threshold x < seq_n i exactly.

E25:Erdős Problem #25 “Must every truncated congruence sieve have logarithmic density?”

E25: “Must every truncated congruence sieve have logarithmic density?”