E66: “Can an additive representation function have a nonzero logarithmic limit?”

Open

Statement	User	Model	Harness	Time
2)V2If a normalized ordered representation function tends to a nonzero constant, then that constant is positive a…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Does there exist a set A of natural numbers for which the ordered additive representation count 1_A*1_A(n), d…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 If a normalized ordered representation function tends to a nonzero constant, then that constant is positive and the underlying set represents every sufficiently large natural as a sum of two elements.

kernel-checked, filed Tue Aug 25 2026 03:53:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every A subset of the naturals and every real c!=0 satisfying the exact logarithmic Tendsto hypothesis from the root.

1) V1 Does there exist a set A of natural numbers for which the ordered additive representation count 1_A*1_A(n), divided by log n, tends to a finite nonzero real constant?

open, filed Tue Aug 25 2026 03:53:12 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Explicit yes-proposition corresponding to the DeepMind answer-placeholder equivalence. Search asymmetry: modern probabilistic search can optimize block-dependent random constructions and Lean can check deterministic consequences, while the no-exceptional-set requirement remains the central obstruction.

Scope. Existential over a set A of natural numbers and a real c!=0; convergence is along natural n tending to infinity, and ordered representations (a,b) with a+b=n are counted.

E66:Erdős Problem #66 “Can an additive representation function have a nonzero logarithmic limit?”

E66: “Can an additive representation function have a nonzero logarithmic limit?”