{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MN44XDMVGJRBSBJOYWGI2QAC5R","short_pith_number":"pith:MN44XDMV","canonical_record":{"source":{"id":"1404.1850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-07T16:51:48Z","cross_cats_sorted":[],"title_canon_sha256":"5087ddf8586bbd4a13f56ab4bf6f40a6bac7b982cdfa581a39930d960e416794","abstract_canon_sha256":"6918d09869f0be3865eb59a2a450f70b2c348790e89c93af89a1ed232f236e7b"},"schema_version":"1.0"},"canonical_sha256":"6379cb8d95326219052ec58c8d4002ec4dca744e3aaf9b1379c61ae0ae3b5cd9","source":{"kind":"arxiv","id":"1404.1850","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.1850","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"arxiv_version","alias_value":"1404.1850v1","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.1850","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"pith_short_12","alias_value":"MN44XDMVGJRB","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MN44XDMVGJRBSBJO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MN44XDMV","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MN44XDMVGJRBSBJOYWGI2QAC5R","target":"record","payload":{"canonical_record":{"source":{"id":"1404.1850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-07T16:51:48Z","cross_cats_sorted":[],"title_canon_sha256":"5087ddf8586bbd4a13f56ab4bf6f40a6bac7b982cdfa581a39930d960e416794","abstract_canon_sha256":"6918d09869f0be3865eb59a2a450f70b2c348790e89c93af89a1ed232f236e7b"},"schema_version":"1.0"},"canonical_sha256":"6379cb8d95326219052ec58c8d4002ec4dca744e3aaf9b1379c61ae0ae3b5cd9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:40.052534Z","signature_b64":"xwsPoo4R/4cUQGUZmzRSi3sBODdSTKgBUbm/2k01bTO90PTgazIIMXmxxzS0zqCf8vf70DfSeR7V3gclvDVWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6379cb8d95326219052ec58c8d4002ec4dca744e3aaf9b1379c61ae0ae3b5cd9","last_reissued_at":"2026-05-18T02:54:40.052150Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:40.052150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1404.1850","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6LPd55Sghf031L0MRGeGjD6UCJ4gPFYE/cokkBwp/WAQwF61U/JO1NJZg/LYQnsxUJsN3KlW4OjhUAaN6kitBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:37:58.440943Z"},"content_sha256":"c28199d4f09aea374c05593e7dc7b91871bef4f566e4d8e356bd3d357fc93e41","schema_version":"1.0","event_id":"sha256:c28199d4f09aea374c05593e7dc7b91871bef4f566e4d8e356bd3d357fc93e41"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MN44XDMVGJRBSBJOYWGI2QAC5R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"\"How many $1$'s are needed?\" revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J. Arias de Reyna, J. van de Lune","submitted_at":"2014-04-07T16:51:48Z","abstract_excerpt":"We present a rigorous and relatively fast method for the computation of the \"complexity\" of a natural number (sequence A005245), and answer some \"old and new\" questions related to the question in the title of this note. We also extend the known terms of the related sequence A005520.\n  We put this paper in the arXiv only for possible reference. This note was written in 2008. It is exactly the copy we have sent to Martin N. Fuller in February 11, 2009 proposing to him to join us as co-author. The ensuing collaboration resulted in a better program and many results about the complexity. We have wa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1850","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:54:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u4rfTaBMIHVBPCZI0ctwYE3+kK8XhCkuLPYwEr9Nx3hKWtsyNmF44+7Lj4gbR5QphkkX9bdkmtn9s+q56lIJAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:37:58.441292Z"},"content_sha256":"ea811e2f3c1adaac19d2e308b79f63f27c51b1d1cbf5f28087215c38b5be130a","schema_version":"1.0","event_id":"sha256:ea811e2f3c1adaac19d2e308b79f63f27c51b1d1cbf5f28087215c38b5be130a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/bundle.json","state_url":"https://pith.science/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T15:37:58Z","links":{"resolver":"https://pith.science/pith/MN44XDMVGJRBSBJOYWGI2QAC5R","bundle":"https://pith.science/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/bundle.json","state":"https://pith.science/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MN44XDMVGJRBSBJOYWGI2QAC5R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MN44XDMVGJRBSBJOYWGI2QAC5R","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":"6918d09869f0be3865eb59a2a450f70b2c348790e89c93af89a1ed232f236e7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-07T16:51:48Z","title_canon_sha256":"5087ddf8586bbd4a13f56ab4bf6f40a6bac7b982cdfa581a39930d960e416794"},"schema_version":"1.0","source":{"id":"1404.1850","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.1850","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"arxiv_version","alias_value":"1404.1850v1","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.1850","created_at":"2026-05-18T02:54:40Z"},{"alias_kind":"pith_short_12","alias_value":"MN44XDMVGJRB","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MN44XDMVGJRBSBJO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MN44XDMV","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:ea811e2f3c1adaac19d2e308b79f63f27c51b1d1cbf5f28087215c38b5be130a","target":"graph","created_at":"2026-05-18T02:54:40Z","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":"We present a rigorous and relatively fast method for the computation of the \"complexity\" of a natural number (sequence A005245), and answer some \"old and new\" questions related to the question in the title of this note. We also extend the known terms of the related sequence A005520.\n  We put this paper in the arXiv only for possible reference. This note was written in 2008. It is exactly the copy we have sent to Martin N. Fuller in February 11, 2009 proposing to him to join us as co-author. The ensuing collaboration resulted in a better program and many results about the complexity. We have wa","authors_text":"J. Arias de Reyna, J. van de Lune","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-07T16:51:48Z","title":"\"How many $1$'s are needed?\" revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1850","kind":"arxiv","version":1},"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:c28199d4f09aea374c05593e7dc7b91871bef4f566e4d8e356bd3d357fc93e41","target":"record","created_at":"2026-05-18T02:54:40Z","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":"6918d09869f0be3865eb59a2a450f70b2c348790e89c93af89a1ed232f236e7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-07T16:51:48Z","title_canon_sha256":"5087ddf8586bbd4a13f56ab4bf6f40a6bac7b982cdfa581a39930d960e416794"},"schema_version":"1.0","source":{"id":"1404.1850","kind":"arxiv","version":1}},"canonical_sha256":"6379cb8d95326219052ec58c8d4002ec4dca744e3aaf9b1379c61ae0ae3b5cd9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6379cb8d95326219052ec58c8d4002ec4dca744e3aaf9b1379c61ae0ae3b5cd9","first_computed_at":"2026-05-18T02:54:40.052150Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:40.052150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xwsPoo4R/4cUQGUZmzRSi3sBODdSTKgBUbm/2k01bTO90PTgazIIMXmxxzS0zqCf8vf70DfSeR7V3gclvDVWDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:40.052534Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.1850","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c28199d4f09aea374c05593e7dc7b91871bef4f566e4d8e356bd3d357fc93e41","sha256:ea811e2f3c1adaac19d2e308b79f63f27c51b1d1cbf5f28087215c38b5be130a"],"state_sha256":"57e1d1885c92bfc4471e69b609405302b4f1110c6a826161c86421d1c469d982"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZOMjavDNBvW47ZA5bNXWay1uQOtICkrzzji0zp5+gWrMou0BUmIeSkouIIaRwlp4KcqhktKvJmAYBE2GqyNLDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:37:58.443264Z","bundle_sha256":"aa245a0e440c646a0b0e75371a55a0ade73b522fd2af811f74019f597b5c6c85"}}