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Here, we show that if $n\\geq k+2$, then $P(F_n^{(k)})>c\\log\\log n$, where $c>0$ is an effectively computable constant. Furthermore, we determine all the $k-$Fibonacci numbers $F_n^{(k)}$ whose largest prime factor is less than or equal to 7."},"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":"1210.4101","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-15T16:51:46Z","cross_cats_sorted":[],"title_canon_sha256":"b3191c24dda73da522a5d0664d370fb0e06f5884e48c2436f2776d82fb8999e8","abstract_canon_sha256":"0cce019e58a9dcde5e60b99bc61836c276294b15a1da45ba23fb286e5f28ce37"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:08.006332Z","signature_b64":"Ofg75MiWWzoD+hrPvmdPbW8L1NdomFZ9+inki+AYGKl7gEdMUu7zCG50rrjvQ2+XaWjsTQ5UwwuocF1I1bOMBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca1d5ea7f38d414883f10128a9806b605614ebced251afebfa81bfd65f476ff8","last_reissued_at":"2026-05-18T03:43:08.005702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:08.005702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the largest prime factor of the $k-$Fibonacci numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Jhon J. Bravo","submitted_at":"2012-10-15T16:51:46Z","abstract_excerpt":"Let $P(m)$ denote the largest prime factor of an integer $m\\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\\geq 2$, let $(F_{n}^{(k)})_{n\\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. Here, we show that if $n\\geq k+2$, then $P(F_n^{(k)})>c\\log\\log n$, where $c>0$ is an effectively computable constant. Furthermore, we determine all the $k-$Fibonacci numbers $F_n^{(k)}$ whose largest prime factor is less than or equal to 7."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4101","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":"1210.4101","created_at":"2026-05-18T03:43:08.005806+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.4101v1","created_at":"2026-05-18T03:43:08.005806+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.4101","created_at":"2026-05-18T03:43:08.005806+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZIOV5J7TRVAU","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZIOV5J7TRVAURA7R","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZIOV5J7T","created_at":"2026-05-18T12:27:30.460161+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/ZIOV5J7TRVAURA7RAEUKTADLMB","json":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB.json","graph_json":"https://pith.science/api/pith-number/ZIOV5J7TRVAURA7RAEUKTADLMB/graph.json","events_json":"https://pith.science/api/pith-number/ZIOV5J7TRVAURA7RAEUKTADLMB/events.json","paper":"https://pith.science/paper/ZIOV5J7T"},"agent_actions":{"view_html":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB","download_json":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB.json","view_paper":"https://pith.science/paper/ZIOV5J7T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.4101&json=true","fetch_graph":"https://pith.science/api/pith-number/ZIOV5J7TRVAURA7RAEUKTADLMB/graph.json","fetch_events":"https://pith.science/api/pith-number/ZIOV5J7TRVAURA7RAEUKTADLMB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB/action/storage_attestation","attest_author":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB/action/author_attestation","sign_citation":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB/action/citation_signature","submit_replication":"https://pith.science/pith/ZIOV5J7TRVAURA7RAEUKTADLMB/action/replication_record"}},"created_at":"2026-05-18T03:43:08.005806+00:00","updated_at":"2026-05-18T03:43:08.005806+00:00"}