{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:XZOCN7LWUDZ5PE4WFYZ5B7A7OY","short_pith_number":"pith:XZOCN7LW","canonical_record":{"source":{"id":"1801.04836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-15T15:05:44Z","cross_cats_sorted":[],"title_canon_sha256":"bea89f0dbd8a9f2b7a9336a25ad77d58ffbe8da8029123df09246a082a6a2d62","abstract_canon_sha256":"54a9acc6d83b4524e30a5168cdda079b60833222636674c7f116f788ffcd64e6"},"schema_version":"1.0"},"canonical_sha256":"be5c26fd76a0f3d793962e33d0fc1f7635e40f5947fdf97625434649679731a7","source":{"kind":"arxiv","id":"1801.04836","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.04836","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1801.04836v1","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.04836","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"XZOCN7LWUDZ5","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"XZOCN7LWUDZ5PE4W","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"XZOCN7LW","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:XZOCN7LWUDZ5PE4WFYZ5B7A7OY","target":"record","payload":{"canonical_record":{"source":{"id":"1801.04836","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-15T15:05:44Z","cross_cats_sorted":[],"title_canon_sha256":"bea89f0dbd8a9f2b7a9336a25ad77d58ffbe8da8029123df09246a082a6a2d62","abstract_canon_sha256":"54a9acc6d83b4524e30a5168cdda079b60833222636674c7f116f788ffcd64e6"},"schema_version":"1.0"},"canonical_sha256":"be5c26fd76a0f3d793962e33d0fc1f7635e40f5947fdf97625434649679731a7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:03.090237Z","signature_b64":"yAjA8qTxmDfxsdpuTpIionYiFViZxT15i1SH7Ba5W+HcqdLvcy3Ooo7pR8pZBxgpbTdq3jp7g/SWn48BKeRLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be5c26fd76a0f3d793962e33d0fc1f7635e40f5947fdf97625434649679731a7","last_reissued_at":"2026-05-18T00:26:03.089570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:03.089570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.04836","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":"2026-05-18T00:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gfoNrYkZA4/CQ9+24LKBDgHZwv2Q8ZkT0b8zn1idAhobLgS/TmyjLaiwrhTKNpCzD5lhnOVi39uiq0vLP50XAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:06:24.656416Z"},"content_sha256":"1cd09c59de855b96e21e17b61f5feb088a1f8666253e924dfaa997baf71bd034","schema_version":"1.0","event_id":"sha256:1cd09c59de855b96e21e17b61f5feb088a1f8666253e924dfaa997baf71bd034"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:XZOCN7LWUDZ5PE4WFYZ5B7A7OY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of representations by a ternary sum of triangular numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Byeong-Kweon Oh, Mingyu Kim","submitted_at":"2018-01-15T15:05:44Z","abstract_excerpt":"For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \\in \\mathbb Z^3$ of $a \\frac{x(x-1)}{2} + b \\frac{y(y-1)}{2} + c \\frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \\cite{s}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04836","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":"2026-05-18T00:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r3jTA031A0ky6ofhq9xsDNovyYDZx/M0uN6ZESuOU60kcz7K2NPfNTkFNBR+xCAYSEZ1tscnT9slDF3RUuEUBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:06:24.656765Z"},"content_sha256":"47e01ee792a8737ec8790abf0dd48b134e97af1a7f4787bf5cfe0bb2a270544f","schema_version":"1.0","event_id":"sha256:47e01ee792a8737ec8790abf0dd48b134e97af1a7f4787bf5cfe0bb2a270544f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/bundle.json","state_url":"https://pith.science/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/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":"2026-06-05T14:06:24Z","links":{"resolver":"https://pith.science/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY","bundle":"https://pith.science/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/bundle.json","state":"https://pith.science/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XZOCN7LWUDZ5PE4WFYZ5B7A7OY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:XZOCN7LWUDZ5PE4WFYZ5B7A7OY","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":"54a9acc6d83b4524e30a5168cdda079b60833222636674c7f116f788ffcd64e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-15T15:05:44Z","title_canon_sha256":"bea89f0dbd8a9f2b7a9336a25ad77d58ffbe8da8029123df09246a082a6a2d62"},"schema_version":"1.0","source":{"id":"1801.04836","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.04836","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1801.04836v1","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.04836","created_at":"2026-05-18T00:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"XZOCN7LWUDZ5","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"XZOCN7LWUDZ5PE4W","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"XZOCN7LW","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:47e01ee792a8737ec8790abf0dd48b134e97af1a7f4787bf5cfe0bb2a270544f","target":"graph","created_at":"2026-05-18T00:26:03Z","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":"For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \\in \\mathbb Z^3$ of $a \\frac{x(x-1)}{2} + b \\frac{y(y-1)}{2} + c \\frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \\cite{s}.","authors_text":"Byeong-Kweon Oh, Mingyu Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-15T15:05:44Z","title":"The number of representations by a ternary sum of triangular numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04836","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":"sha256:1cd09c59de855b96e21e17b61f5feb088a1f8666253e924dfaa997baf71bd034","target":"record","created_at":"2026-05-18T00:26:03Z","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":"54a9acc6d83b4524e30a5168cdda079b60833222636674c7f116f788ffcd64e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-15T15:05:44Z","title_canon_sha256":"bea89f0dbd8a9f2b7a9336a25ad77d58ffbe8da8029123df09246a082a6a2d62"},"schema_version":"1.0","source":{"id":"1801.04836","kind":"arxiv","version":1}},"canonical_sha256":"be5c26fd76a0f3d793962e33d0fc1f7635e40f5947fdf97625434649679731a7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be5c26fd76a0f3d793962e33d0fc1f7635e40f5947fdf97625434649679731a7","first_computed_at":"2026-05-18T00:26:03.089570Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:03.089570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yAjA8qTxmDfxsdpuTpIionYiFViZxT15i1SH7Ba5W+HcqdLvcy3Ooo7pR8pZBxgpbTdq3jp7g/SWn48BKeRLAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:03.090237Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.04836","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cd09c59de855b96e21e17b61f5feb088a1f8666253e924dfaa997baf71bd034","sha256:47e01ee792a8737ec8790abf0dd48b134e97af1a7f4787bf5cfe0bb2a270544f"],"state_sha256":"8c63f88f3db50e7e549c514407771740250bbace60d4af7bb301ce3c82fcb6da"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IgfM3WUEpMvR9QAtRYlR3mzw+QQAmBhBR3RpEMFupZnRKh18PZ14mDoQMA+y8RvNWsNLgpNaZy+kjP6jegA6Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T14:06:24.658607Z","bundle_sha256":"00a95f6cc6683a5a875cfe1b8fa335258790acd0feb94d32760521551d3bb461"}}