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Later, Anick and Ramras \\cite{AR2013} showed that these two conditions are also sufficient for odd $n \\leq 2^{32}$ and conjectured that this was true for all odd $n$. In this note we prove the conjecture."},"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":"1310.6776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-24T21:20:13Z","cross_cats_sorted":[],"title_canon_sha256":"5ae7e80101537c301e8c6f5b18aa2a490da8916c0563f20cc566245e563cc540","abstract_canon_sha256":"27e291f6f8ec8ca0d127f14964cdbcb3d23ed183bad8db370f9e1bc762e2d2d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:57.614542Z","signature_b64":"HxXIcS2CYbzSvXEcGHI5rH526uEOKUsmJOibIY9MED6/hdQHxwOZY/L6aPJMm4M/NPD00W36ISU8aCpn95TADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e525dd843196e0a6157dbd287b1f0e3732647ac4521ec6b2b6a581fd3eadca2d","last_reissued_at":"2026-05-18T03:08:57.613628Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:57.613628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposing the cube into paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joshua Erde","submitted_at":"2013-10-24T21:20:13Z","abstract_excerpt":"We consider the question of when the $n$-dimensional hypercube can be decomposed into paths of length $k$. 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