{"paper":{"title":"On a class of Lebesgue-Ramanujan-Nagell equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Azizul Hoque","submitted_at":"2020-05-11T16:03:21Z","abstract_excerpt":"We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\\geq 1, m\\geq 0$ and $n\\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\\not\\equiv 3 \\pmod 4$. We first solve this equation for prime $n$, under the condition $n\\nmid h(-cd)$, where $h(-cd)$ denotes the class number of the quadratic field $\\mathbb{Q}(\\sqrt{-cd})$. We then completely solve this equation for both $c$ and $d$ primes under the assumption that $\\gcd(n, h(-cd))=1$. We also completely solve this equation for $c=1$ and $d\\equiv1 \\pmod 4$, under the condition $\\gcd(n, h(-d))="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.05214","kind":"arxiv","version":3},"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/2005.05214/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"}