{"paper":{"title":"Upper bounds on the solutions to $n = p+m^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aran Nayebi","submitted_at":"2010-04-04T21:54:52Z","abstract_excerpt":"Hardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $\\mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by \\mathcal{R}(n) \\sim \\frac{\\sqrt{n}}{\\log n}\\prod_{p=3}^{\\infty}(1-\\frac{1}{p-1}(\\frac{n}{p})), where $p$ is a prime, $m$ is an integer, and $(\\frac{n}{p})$ denotes the Legendre symbol. Unfortunately, as we will later point out, this conjecture is difficult to prove and not \\emph{all} integers that are nonsquares can be represented as the sum of a prime and a square"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0536","kind":"arxiv","version":7},"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"}