{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:PDK2TWNXOOLRT75VR6KQBA7FGA","short_pith_number":"pith:PDK2TWNX","schema_version":"1.0","canonical_sha256":"78d5a9d9b7739719ffb58f950083e5302dde91c24e26e85b74a27cac0f00b110","source":{"kind":"arxiv","id":"2506.14379","version":4},"attestation_state":"computed","paper":{"title":"On a Diophantine Equation Involving Lucas Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nazim I. Mahmudov, Seyran S. Ibrahimov","submitted_at":"2025-06-17T10:24:32Z","abstract_excerpt":"Let L_t denote the t-th Lucas number. We prove that the Diophantine equation\n  L_m^{n+k} + L_m^n = L_r\n  has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof."},"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":"2506.14379","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-17T10:24:32Z","cross_cats_sorted":[],"title_canon_sha256":"3e1363b48ecabcd68f5c2b618ff8f551d3ef8ca133bd58cd871ece87334cae08","abstract_canon_sha256":"a1d6fba1ddb6c922ef9ee1909bd5344cc55117cf2ccb0327776db0149d153b8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:20:44.548607Z","signature_b64":"lH/WbC3cO4zb6+d5YB/AU+c+fuj7y+gYs9CaHidkZeaDLRuEtyWF6Vo0qxb/2xCQL4cIyyU9t6BXv4TErIx4AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78d5a9d9b7739719ffb58f950083e5302dde91c24e26e85b74a27cac0f00b110","last_reissued_at":"2026-07-14T01:20:44.547597Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:20:44.547597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a Diophantine Equation Involving Lucas Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nazim I. Mahmudov, Seyran S. Ibrahimov","submitted_at":"2025-06-17T10:24:32Z","abstract_excerpt":"Let L_t denote the t-th Lucas number. We prove that the Diophantine equation\n  L_m^{n+k} + L_m^n = L_r\n  has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.14379","kind":"arxiv","version":4},"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/2506.14379/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.14379","created_at":"2026-07-14T01:20:44.548065+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.14379v4","created_at":"2026-07-14T01:20:44.548065+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.14379","created_at":"2026-07-14T01:20:44.548065+00:00"},{"alias_kind":"pith_short_12","alias_value":"PDK2TWNXOOLR","created_at":"2026-07-14T01:20:44.548065+00:00"},{"alias_kind":"pith_short_16","alias_value":"PDK2TWNXOOLRT75V","created_at":"2026-07-14T01:20:44.548065+00:00"},{"alias_kind":"pith_short_8","alias_value":"PDK2TWNX","created_at":"2026-07-14T01:20:44.548065+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/PDK2TWNXOOLRT75VR6KQBA7FGA","json":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA.json","graph_json":"https://pith.science/api/pith-number/PDK2TWNXOOLRT75VR6KQBA7FGA/graph.json","events_json":"https://pith.science/api/pith-number/PDK2TWNXOOLRT75VR6KQBA7FGA/events.json","paper":"https://pith.science/paper/PDK2TWNX"},"agent_actions":{"view_html":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA","download_json":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA.json","view_paper":"https://pith.science/paper/PDK2TWNX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.14379&json=true","fetch_graph":"https://pith.science/api/pith-number/PDK2TWNXOOLRT75VR6KQBA7FGA/graph.json","fetch_events":"https://pith.science/api/pith-number/PDK2TWNXOOLRT75VR6KQBA7FGA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA/action/storage_attestation","attest_author":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA/action/author_attestation","sign_citation":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA/action/citation_signature","submit_replication":"https://pith.science/pith/PDK2TWNXOOLRT75VR6KQBA7FGA/action/replication_record"}},"created_at":"2026-07-14T01:20:44.548065+00:00","updated_at":"2026-07-14T01:20:44.548065+00:00"}