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The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $\\phi n$ where $\\phi=(1+\\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$"},"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":"1903.08274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-03-19T22:23:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"af3809d5489a290fa2926e173c1ad98843de86ace7c208b91a2f5df9b4261f01","abstract_canon_sha256":"28dfa25e9ecf7d363009d84ea2439ef74e21ef7724a404a153c738c458a3dec8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:49.827231Z","signature_b64":"SjvzRqD6awYKJACwc63l9zbEFGwRN2Ovh3h5eUDHm+tvgtcdV3lYQGGuDbXRHkiGyfdQGPdZupSQkmegew3QAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43cf466a765fe73008f8dc7a37fa5ae8335fb196b5891942c60254e86b9a1d5e","last_reissued_at":"2026-05-17T23:50:49.826557Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:49.826557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Slow Fibonacci Walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Fan Chung, Ron Graham, Sam Spiro","submitted_at":"2019-03-19T22:23:26Z","abstract_excerpt":"For a positive integer $n$, we study the number of steps to reach $n$ by a {\\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. 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