{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5NUNQNH27PRFN5GHXVPGYB4M5L","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"91a97af14caf6595908d0187fbaaff7a7260b3e6cd996f48533de4772b6dc069","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-08T14:05:28Z","title_canon_sha256":"2f8106aaffbb11131d03224a0757b7af44fd09cf18524e01049620c6fb84406c"},"schema_version":"1.0","source":{"id":"1407.2091","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.2091","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"arxiv_version","alias_value":"1407.2091v3","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.2091","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"pith_short_12","alias_value":"5NUNQNH27PRF","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5NUNQNH27PRFN5GH","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5NUNQNH2","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:291dba399b9e9ded8dd340f49845d67963f0712718c90152f014d9930bb66df0","target":"graph","created_at":"2026-05-18T00:24:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The polynomial $X^{3}-X-1$ has a unique positive root known as plastic number, which is denoted by $\\rho$ and is approximately equal to $1.32471795$. In this note we study the zeroes of the generalized polynomial $X^{k}-\\sum_{j=0}^{k-2}X^{j}$ for $k\\geq 3$ and prove that its unique positive root $\\lambda_{k}$ tends to the golden ratio $\\phi=\\frac{1+\\sqrt{5}}{2}$ as $k \\to \\infty$. We also derive bounds on $\\lambda_{k}$ in terms of Fibonacci numbers.","authors_text":"Vasileios Iliopoulos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-08T14:05:28Z","title":"The plastic number and its generalized polynomial"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2091","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:18a36abbff5ae3d994d6940017b56a7e21c1d809a147743054ca3178e2a37771","target":"record","created_at":"2026-05-18T00:24:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"91a97af14caf6595908d0187fbaaff7a7260b3e6cd996f48533de4772b6dc069","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-08T14:05:28Z","title_canon_sha256":"2f8106aaffbb11131d03224a0757b7af44fd09cf18524e01049620c6fb84406c"},"schema_version":"1.0","source":{"id":"1407.2091","kind":"arxiv","version":3}},"canonical_sha256":"eb68d834fafbe256f4c7bd5e6c078cead5976762913211e99cce2aefba84878e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb68d834fafbe256f4c7bd5e6c078cead5976762913211e99cce2aefba84878e","first_computed_at":"2026-05-18T00:24:32.150350Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:32.150350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WcSZVJNyI6TF8YgZPlCJUVDyAMhuU3Ctqp85dXxvJI8zeKDYQq7aCopO1kDcMJfIzXgslnmknetHT0P0BGvACA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:32.150842Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.2091","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18a36abbff5ae3d994d6940017b56a7e21c1d809a147743054ca3178e2a37771","sha256:291dba399b9e9ded8dd340f49845d67963f0712718c90152f014d9930bb66df0"],"state_sha256":"9eb49b8feadb153d82816af1b12dca135d3285aa2f087ec4dd02868827e3e4ea"}