{"paper":{"title":"On the $x-$coordinates of Pell equations which are sums of two Padovan numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mahadi Ddamulira","submitted_at":"2019-05-27T16:15:42Z","abstract_excerpt":"Let $ \\{P_{n}\\}_{n\\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\\geq 0 $. In this paper, we find all positive square-free integers $ d $ such that the Pell equations $ x^2-dy^2 = \\pm 1 $, $ X^2-dY^2=\\pm 4 $ have at least two positive integer solutions $ (x,y) $ and $(x^{\\prime}, y^{\\prime})$, $ (X,Y) $ and $(X^{\\prime}, Y^{\\prime})$, respectively, such that each of $ x, ~x^{\\prime}, ~X, ~X^{\\prime} $ is a sum of two Padovan numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.11322","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":""},"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"}