{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:3PM26LJQLKXYDPFPSAJUYWZN3A","short_pith_number":"pith:3PM26LJQ","canonical_record":{"source":{"id":"2001.05325","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","cross_cats_sorted":[],"title_canon_sha256":"2aba0bae5653b71ad30b01cb88ed0dc4e90b2f185fb9951d7a7efaec98674185","abstract_canon_sha256":"aecadab99e3b9197e86520c7d30987e08df201fceae3cdd77178db76d0690cb2"},"schema_version":"1.0"},"canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","source":{"kind":"arxiv","id":"2001.05325","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2001.05325","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"arxiv_version","alias_value":"2001.05325v4","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.05325","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_12","alias_value":"3PM26LJQLKXY","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_16","alias_value":"3PM26LJQLKXYDPFP","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_8","alias_value":"3PM26LJQ","created_at":"2026-07-05T00:51:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:3PM26LJQLKXYDPFPSAJUYWZN3A","target":"record","payload":{"canonical_record":{"source":{"id":"2001.05325","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","cross_cats_sorted":[],"title_canon_sha256":"2aba0bae5653b71ad30b01cb88ed0dc4e90b2f185fb9951d7a7efaec98674185","abstract_canon_sha256":"aecadab99e3b9197e86520c7d30987e08df201fceae3cdd77178db76d0690cb2"},"schema_version":"1.0"},"canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:51:38.348482Z","signature_b64":"fS/A2lHvu+xnfGZ7XPA/B66VkwFs/PnO1laot89WtUBabGMWMDk9yCEBRe/mQVoU6ney0QRLYa8xL2jLo5edAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","last_reissued_at":"2026-07-05T00:51:38.347944Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:51:38.347944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2001.05325","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:51:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LO9XbgGLC3OURVB0tjd3Xw1CEyMeUIzPPGgYZ/t/RpQnyPrMCPcXWBXlrzOhH1e+vqowa8P5i5MYypZLhfVeCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T00:00:05.397381Z"},"content_sha256":"ce32c05a5b00f6a2fbe518e1e6e8942a8d4780ccec204c19a743654df2b148ca","schema_version":"1.0","event_id":"sha256:ce32c05a5b00f6a2fbe518e1e6e8942a8d4780ccec204c19a743654df2b148ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:3PM26LJQLKXYDPFPSAJUYWZN3A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On sums of four pentagonal numbers with coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitry Krachun, Zhi-Wei Sun","submitted_at":"2020-01-14T14:59:35Z","abstract_excerpt":"The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\\ (n=0,1,2,\\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.05325","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.05325/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:51:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mr7HLugLUcLkyEN5rzfmUd7fPbMj67CM+CpYWCTE6Izw6ug7A9eM3wlVYTGBSP9YNOTvzp9Y1fD5/RY308dPCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T00:00:05.398062Z"},"content_sha256":"98908d60d111e46ad4ccac5277ce4a6aeb920498a9fb195a6bf5ec9c6e92e03c","schema_version":"1.0","event_id":"sha256:98908d60d111e46ad4ccac5277ce4a6aeb920498a9fb195a6bf5ec9c6e92e03c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/bundle.json","state_url":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/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":"2026-08-04T00:00:05Z","links":{"resolver":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A","bundle":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/bundle.json","state":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:3PM26LJQLKXYDPFPSAJUYWZN3A","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":"aecadab99e3b9197e86520c7d30987e08df201fceae3cdd77178db76d0690cb2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","title_canon_sha256":"2aba0bae5653b71ad30b01cb88ed0dc4e90b2f185fb9951d7a7efaec98674185"},"schema_version":"1.0","source":{"id":"2001.05325","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2001.05325","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"arxiv_version","alias_value":"2001.05325v4","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.05325","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_12","alias_value":"3PM26LJQLKXY","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_16","alias_value":"3PM26LJQLKXYDPFP","created_at":"2026-07-05T00:51:38Z"},{"alias_kind":"pith_short_8","alias_value":"3PM26LJQ","created_at":"2026-07-05T00:51:38Z"}],"graph_snapshots":[{"event_id":"sha256:98908d60d111e46ad4ccac5277ce4a6aeb920498a9fb195a6bf5ec9c6e92e03c","target":"graph","created_at":"2026-07-05T00:51:38Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2001.05325/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\\ (n=0,1,2,\\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.","authors_text":"Dmitry Krachun, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","title":"On sums of four pentagonal numbers with coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.05325","kind":"arxiv","version":4},"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:ce32c05a5b00f6a2fbe518e1e6e8942a8d4780ccec204c19a743654df2b148ca","target":"record","created_at":"2026-07-05T00:51:38Z","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":"aecadab99e3b9197e86520c7d30987e08df201fceae3cdd77178db76d0690cb2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","title_canon_sha256":"2aba0bae5653b71ad30b01cb88ed0dc4e90b2f185fb9951d7a7efaec98674185"},"schema_version":"1.0","source":{"id":"2001.05325","kind":"arxiv","version":4}},"canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","first_computed_at":"2026-07-05T00:51:38.347944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:51:38.347944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fS/A2lHvu+xnfGZ7XPA/B66VkwFs/PnO1laot89WtUBabGMWMDk9yCEBRe/mQVoU6ney0QRLYa8xL2jLo5edAw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:51:38.348482Z","signed_message":"canonical_sha256_bytes"},"source_id":"2001.05325","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce32c05a5b00f6a2fbe518e1e6e8942a8d4780ccec204c19a743654df2b148ca","sha256:98908d60d111e46ad4ccac5277ce4a6aeb920498a9fb195a6bf5ec9c6e92e03c"],"state_sha256":"c1cb09535bc40554cd2a93b7fadbebf8fec95f5710d2d666882850c5a0fe61f9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yCrEN4rgewF2vHK9BW1TIjHh05wsVaiDQzKXVmwyPjGjBDPpE92rrlxrMWPYXm0gjaG5TciYFiks1NRyPNBWCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T00:00:05.404095Z","bundle_sha256":"d6f9fdf4fcd2679f3291ae71b363edcc46227ee56f8c73f73412fd32fae7f6b7"}}