{"paper":{"title":"On integers as the sum of a prime and a $k$-th power","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.NT","authors_text":"Aran Nayebi","submitted_at":"2009-08-05T18:33:44Z","abstract_excerpt":"Let $\\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\\{n \\le X, n \\in I_k, n\\text{not a sum of a prime and a $k$-th power}\\}|.\n  Hardy and Littlewood conjectured that for $k = 2$ and $k=3$, E_k(X) \\ll_{k} 1. In this note we present an alternative approach grounded in the theory of Diophantine equations towards a proof of the conjecture for all $k \\ge 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0554","kind":"arxiv","version":25},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}