{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:DX5YOWOOTHPEDO3BLREZDRU2PH","short_pith_number":"pith:DX5YOWOO","canonical_record":{"source":{"id":"0812.1724","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-09T16:39:49Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cbba0471dbb1d019e1ed1f76c6dcd4f31c7e55c93dad28af1027735328552e29","abstract_canon_sha256":"949b9a0dfeadca05f30ca4b7d6f8fc1816c71db36bd8fa703eaec2fd1e6d6820"},"schema_version":"1.0"},"canonical_sha256":"1dfb8759ce99de41bb615c4991c69a79ea06b602a8715fb25917d28fccd90aa8","source":{"kind":"arxiv","id":"0812.1724","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.1724","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"arxiv_version","alias_value":"0812.1724v3","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1724","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"pith_short_12","alias_value":"DX5YOWOOTHPE","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"DX5YOWOOTHPEDO3B","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"DX5YOWOO","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:DX5YOWOOTHPEDO3BLREZDRU2PH","target":"record","payload":{"canonical_record":{"source":{"id":"0812.1724","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-09T16:39:49Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cbba0471dbb1d019e1ed1f76c6dcd4f31c7e55c93dad28af1027735328552e29","abstract_canon_sha256":"949b9a0dfeadca05f30ca4b7d6f8fc1816c71db36bd8fa703eaec2fd1e6d6820"},"schema_version":"1.0"},"canonical_sha256":"1dfb8759ce99de41bb615c4991c69a79ea06b602a8715fb25917d28fccd90aa8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:11.238990Z","signature_b64":"5v/yFBEm7PPsES7vI8Vph2t+rJN/jfnEIeHEbWF7g0RSAWA/zM/xEIjVME28lVcqI+OyKVIDS+2/M7n7kMt1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1dfb8759ce99de41bb615c4991c69a79ea06b602a8715fb25917d28fccd90aa8","last_reissued_at":"2026-05-18T02:15:11.238201Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:11.238201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0812.1724","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:15:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yzwvcPpQ2zG/37Fy8Nhb1Io3LLK7DKku9SOVQFbxJZD+1atQXJmnVfXVuScjsC/WIj8os2LxJKwarAmKWpEhCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T10:45:03.897383Z"},"content_sha256":"69b272acc225c76e9ee67beac9d451c05ae3689559bf7e572e98b9317af0fc32","schema_version":"1.0","event_id":"sha256:69b272acc225c76e9ee67beac9d451c05ae3689559bf7e572e98b9317af0fc32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:DX5YOWOOTHPEDO3BLREZDRU2PH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Bialostocki's conjecture for zero-sum sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Song Guo, Zhi-Wei Sun","submitted_at":"2008-12-09T16:39:49Z","abstract_excerpt":"Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\\sum_{k=1}^n a_k\\equiv\\sum_{k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $\\sigma$ on {1,...,n} such that $\\sum_{k=1}^n w_k a_{\\sigma(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1724","kind":"arxiv","version":3},"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":"2026-05-18T02:15:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tSSQ7mOFOkf1h9RUESzCIxqk8kWcxt1LGnhL6i+38lVdHPm2cS8JNFntPwEpBwInJtpNVjIBzPu7dzlccIZWAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T10:45:03.898101Z"},"content_sha256":"828dea65596b666f777c1a440cd92c79eb82e3fa0d0192d66e129d42af1e5a95","schema_version":"1.0","event_id":"sha256:828dea65596b666f777c1a440cd92c79eb82e3fa0d0192d66e129d42af1e5a95"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/bundle.json","state_url":"https://pith.science/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/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":"2026-05-23T10:45:03Z","links":{"resolver":"https://pith.science/pith/DX5YOWOOTHPEDO3BLREZDRU2PH","bundle":"https://pith.science/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/bundle.json","state":"https://pith.science/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DX5YOWOOTHPEDO3BLREZDRU2PH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:DX5YOWOOTHPEDO3BLREZDRU2PH","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":"949b9a0dfeadca05f30ca4b7d6f8fc1816c71db36bd8fa703eaec2fd1e6d6820","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-09T16:39:49Z","title_canon_sha256":"cbba0471dbb1d019e1ed1f76c6dcd4f31c7e55c93dad28af1027735328552e29"},"schema_version":"1.0","source":{"id":"0812.1724","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.1724","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"arxiv_version","alias_value":"0812.1724v3","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1724","created_at":"2026-05-18T02:15:11Z"},{"alias_kind":"pith_short_12","alias_value":"DX5YOWOOTHPE","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"DX5YOWOOTHPEDO3B","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"DX5YOWOO","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:828dea65596b666f777c1a440cd92c79eb82e3fa0d0192d66e129d42af1e5a95","target":"graph","created_at":"2026-05-18T02:15:11Z","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":"Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\\sum_{k=1}^n a_k\\equiv\\sum_{k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $\\sigma$ on {1,...,n} such that $\\sum_{k=1}^n w_k a_{\\sigma(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference.","authors_text":"Song Guo, Zhi-Wei Sun","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-09T16:39:49Z","title":"On Bialostocki's conjecture for zero-sum sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1724","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":"sha256:69b272acc225c76e9ee67beac9d451c05ae3689559bf7e572e98b9317af0fc32","target":"record","created_at":"2026-05-18T02:15:11Z","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":"949b9a0dfeadca05f30ca4b7d6f8fc1816c71db36bd8fa703eaec2fd1e6d6820","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-09T16:39:49Z","title_canon_sha256":"cbba0471dbb1d019e1ed1f76c6dcd4f31c7e55c93dad28af1027735328552e29"},"schema_version":"1.0","source":{"id":"0812.1724","kind":"arxiv","version":3}},"canonical_sha256":"1dfb8759ce99de41bb615c4991c69a79ea06b602a8715fb25917d28fccd90aa8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1dfb8759ce99de41bb615c4991c69a79ea06b602a8715fb25917d28fccd90aa8","first_computed_at":"2026-05-18T02:15:11.238201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:11.238201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5v/yFBEm7PPsES7vI8Vph2t+rJN/jfnEIeHEbWF7g0RSAWA/zM/xEIjVME28lVcqI+OyKVIDS+2/M7n7kMt1BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:11.238990Z","signed_message":"canonical_sha256_bytes"},"source_id":"0812.1724","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69b272acc225c76e9ee67beac9d451c05ae3689559bf7e572e98b9317af0fc32","sha256:828dea65596b666f777c1a440cd92c79eb82e3fa0d0192d66e129d42af1e5a95"],"state_sha256":"b92d6d7cd0afc44c37f8bb02d4a5d93d860c3985aa16d249cf9812e20c33e104"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hEThNNnAZuU2QtboMZh1NeYM/whsYgq98pAyymvxAZnuk6O4AAPXXQYsacp+AbazR3/tOKEIKNx3UaKBiilODw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T10:45:03.902172Z","bundle_sha256":"c523ed655fdd8c431362279ace78ac894551b093ab3ea843f0f064845a541486"}}