{"paper":{"title":"On the factorization of $x^2+D$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Amir Ghadermarzi","submitted_at":"2017-09-09T13:44:34Z","abstract_excerpt":"Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \\nmid D$, and $0< \\sigma < 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,\\sigma)}$, then there exists an effectively computable constant $C_p$ such that for every $x> C_P$ with $x^2+D=p^n.m $, we have $ m > x^{\\sigma}$. As an application, we show that for $x \\neq \\{1015,5 \\}$, if the equation $x^2+76=101^n.m $ holds, we have $ m > x^{0.14}$. ."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02954","kind":"arxiv","version":2},"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"}