{"paper":{"title":"Theorem $(1+1.9)$ on the Goldbach Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jiamin Li, Jianya Liu","submitted_at":"2026-06-01T15:03:44Z","abstract_excerpt":"For $1 \\leq a \\leq 2$, we say Proposition $(1+a)$ holds if every sufficiently large even integer $N$ can be written as $$N = p + rq, \\quad r \\leq q^{a-1},$$ where $r$ is either $1$ or prime, and $p,q$ are primes. Thus Proposition $(1+1)$ is essentially the binary Goldbach Conjecture, and Proposition $(1+2)$ is Chen's theorem. We prove unconditionally that Proposition $(1+1.9)$ is true. Assuming the Elliott--Halberstam Conjecture, the exponent $1.9$ can be improved to $1.4$. Analogously, Proposition $(1-a)$ is formulated for the Twin Prime Conjecture. Unconditionally, we prove Proposition $(1-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05224","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05224/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}