{"data":{"id":"erdos-turan-basis-1941","statement_oneline":"If every large integer is a sum of two elements of a set B, the number of such representations must be unbounded.","statement_full_latex":"If $B \\subseteq \\mathbb{N}$ is an additive basis of order 2 (every sufficiently large $n$ is $b_1 + b_2$ with $b_i \\in B$), then the representation function $r_B(n)$ is unbounded (Erdős-Turán, 1941).","source":{"title":"Erdős-Turán conjecture on additive bases","arxiv":null,"url":"https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n_conjecture_on_additive_bases","year":1941,"area":"math.NT"},"renown":{"score":3,"rationale":"A cornerstone of additive combinatorics, open for 85 years.","scored_by":"founder"},"attackability":{"score":1,"rationale":"A counterexample is an infinite set with bounded representation function -- not a finite witness, though a strong finite pattern with a provable extension rule could in principle certify one.","subscores":{"finite_witness":1,"oracle_cost":2,"freshness":1,"seedability":1},"scored_by":"founder"},"status":"open","last_verified_open":"2026-01-15","added":"2026-07-25","ranking_score":3,"verification_tier":null,"ai_systems":[],"posed_by":null,"years_open":null,"notability":null,"source_trackers":[],"resolution_type":null,"resolver":null,"resolution_date":null,"evidence_url":null,"evidence_note":null,"method_note":null,"effective_status":"open","claims":[],"status_events":[{"id":"seed-erdos-turan-basis-1941","conjecture_id":"erdos-turan-basis-1941","status":"open","date":"2026-07-25T00:00:00.000Z","actor":"human-admin","rationale":"Loaded from unified seed data.","note":"Loaded from unified seed data."}],"public_actions":[{"name":"claim","method":"POST","href":"/c/erdos-turan-basis-1941/claim","verification":"email magic link"},{"name":"report_resolution","method":"POST","href":"/c/erdos-turan-basis-1941/resolve","verification":"admin review"}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}