Erdős Problem #330 — A question of Erdös and Nathanson is the following.
A question of Erdös and Nathanson is the following. Suppose is a minimal basis which has positive density. Can it happen that for any , the (upper) density of the integers which cannot be represented without using is positive?
References
Additional references
Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28 (1980).
Progress summary
A construction has been claimed, but no independently verified proof is available, so the question remains open.
The problem asks whether a minimal additive basis of positive density can be so sensitive that deleting any leaves a positive-density set of integers unrepresentable. It was posed by Erdős and Nathanson; the intended density convention for may have been positive upper density.
Known results
- Nathanson (1987) records earlier minimal-basis work by Stöhr, Härtter, and Nathanson, while identifying the positive-density strengthening as open.
Recent AI and formalization claims
A construction was claimed after work involving GPT 5.5 Pro, with Codex used in formalization; a separate claim credits Seed-Prover 1.5. A formalization issue reports a theorem for an order- variant with positive upper density, but neither claim has an independently verified published proof.
Current status (as of March 2026): The problem has claimed constructions, including an order- formalization variant, but no verified proof has been established; the original question remains open.
Solutions 0
No solutions have been posted yet.