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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.11322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-05-27T16:15:42Z","cross_cats_sorted":[],"title_canon_sha256":"15c872b66e4c0034cd3e071ea5eebf4898dcd1fb3012e5badd51a14836f0f69d","abstract_canon_sha256":"8b62caa6351aebf03cc99e02e591f61a279d448e765d6cafbd29431a3eb5d929"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:00.706494Z","signature_b64":"Eg6RXRMpGVTojQWYWkJMK2E1X5rJfjQUPPF9dkBaoAj6Lzw1qMgxhiCmJ80sPq84ik+PkAQw+AjEtBtki5FNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a59b274ef48599e10985fd0beb9767ada4046e0c9a8c0bc2f6a7509a6ba9c94","last_reissued_at":"2026-05-17T23:45:00.705751Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:00.705751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1905.11322","created_at":"2026-05-17T23:45:00.705887+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.11322v1","created_at":"2026-05-17T23:45:00.705887+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.11322","created_at":"2026-05-17T23:45:00.705887+00:00"},{"alias_kind":"pith_short_12","alias_value":"BJM3E5HPJBMZ","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"BJM3E5HPJBMZ4EEY","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"BJM3E5HP","created_at":"2026-05-18T12:33:12.712433+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL","json":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL.json","graph_json":"https://pith.science/api/pith-number/BJM3E5HPJBMZ4EEYL7IL5OLWPL/graph.json","events_json":"https://pith.science/api/pith-number/BJM3E5HPJBMZ4EEYL7IL5OLWPL/events.json","paper":"https://pith.science/paper/BJM3E5HP"},"agent_actions":{"view_html":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL","download_json":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL.json","view_paper":"https://pith.science/paper/BJM3E5HP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.11322&json=true","fetch_graph":"https://pith.science/api/pith-number/BJM3E5HPJBMZ4EEYL7IL5OLWPL/graph.json","fetch_events":"https://pith.science/api/pith-number/BJM3E5HPJBMZ4EEYL7IL5OLWPL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL/action/storage_attestation","attest_author":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL/action/author_attestation","sign_citation":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL/action/citation_signature","submit_replication":"https://pith.science/pith/BJM3E5HPJBMZ4EEYL7IL5OLWPL/action/replication_record"}},"created_at":"2026-05-17T23:45:00.705887+00:00","updated_at":"2026-05-17T23:45:00.705887+00:00"}