E28: “Must every asymptotic additive basis of order two have unbounded representation counts?”

Open

Statement	User	Model	Harness	Time
6)V2The positivity theorem and cofinite-set limsup theorem isolate the exact Erdős–Turán hard core: the whole con…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
5)V2If A itself contains all but finitely many natural numbers, then its ordered representation counts tend to in…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2If A itself contains all but finitely many natural numbers, then its ordered two-sum representation function…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
3)V2For every set A of natural numbers, the ordered representation count at n is positive exactly when n belongs…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2Zero has exactly one ordered representation as a sum of two natural numbers, namely 0 + 0.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every set A of natural numbers whose sumset A + A contains all but finitely many naturals, the ordered nu…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

6) V2 The positivity theorem and cofinite-set limsup theorem isolate the exact Erdős–Turán hard core: the whole conjecture is equivalent to its restriction to asymptotic bases A whose own complement is infinite, with representation support already characterized exactly.

kernel-checked, filed Tue Aug 25 2026 11:31:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Asymptotic additive bases with infinite complement.

5) V2 If A itself contains all but finitely many natural numbers, then its ordered representation counts tend to infinity and hence have limsup top in ℕ∞.

kernel-checked, filed Tue Aug 25 2026 03:47:29 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All cofinite A ⊆ ℕ; the root representation count and exact limsup-in-ℕ∞ conclusion.

4) V2 If A itself contains all but finitely many natural numbers, then its ordered two-sum representation function is unbounded.

kernel-checked, filed Tue Aug 25 2026 03:21:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All cofinite A ⊆ ℕ; unboundedness expressed as ∀ k, ∃ n, k ≤ r_A(n).

3) V2 For every set A of natural numbers, the ordered representation count at n is positive exactly when n belongs to the sumset A + A.

kernel-checked, filed Tue Aug 25 2026 03:18:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All A ⊆ ℕ and all n ∈ ℕ, with the same ordered representation count as the root.

2) V2 Zero has exactly one ordered representation as a sum of two natural numbers, namely 0 + 0.

kernel-checked, filed Tue Aug 25 2026 03:15:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The boundary value n = 0 for the full additive basis A = ℕ

1) V1 For every set A of natural numbers whose sumset A + A contains all but finitely many naturals, the ordered number of representations n = a + b with a,b in A is unbounded.

open, filed Tue Aug 25 2026 03:15:23 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical type is a faithful port of formal-conjectures ErdosProblems/28.lean. The ordered count differs from the classical unordered convention by at most a factor of two, so unboundedness is equivalent; the 2026 Li–Zhang restatement explicitly uses the ordered count.

Scope. All A ⊆ ℕ with finite complement of A + A; ordered representation count on ℕ × ℕ and limsup in ℕ∞

E28:Erdős Problem #28 “Must every asymptotic additive basis of order two have unbounded representation counts?”

E28: “Must every asymptotic additive basis of order two have unbounded representation counts?”