{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:DNQYVWRX6NCJYKZUINDAK7MR4H","short_pith_number":"pith:DNQYVWRX","canonical_record":{"source":{"id":"1904.06369","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-12T18:27:09Z","cross_cats_sorted":[],"title_canon_sha256":"37fd0487741fbb3fe9d145952fab4a40d1aa4c6698cbcc9bfe163de612a851dd","abstract_canon_sha256":"f2cefb4f3981336290b7df8c9e095dd38768bc300befd212d73ff42e0b44c9f9"},"schema_version":"1.0"},"canonical_sha256":"1b618ada37f3449c2b344346057d91e1e43a5e53c37c40698ed407aa14d62573","source":{"kind":"arxiv","id":"1904.06369","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06369","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06369v3","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06369","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"pith_short_12","alias_value":"DNQYVWRX6NCJ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"DNQYVWRX6NCJYKZU","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"DNQYVWRX","created_at":"2026-05-18T12:33:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:DNQYVWRX6NCJYKZUINDAK7MR4H","target":"record","payload":{"canonical_record":{"source":{"id":"1904.06369","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-12T18:27:09Z","cross_cats_sorted":[],"title_canon_sha256":"37fd0487741fbb3fe9d145952fab4a40d1aa4c6698cbcc9bfe163de612a851dd","abstract_canon_sha256":"f2cefb4f3981336290b7df8c9e095dd38768bc300befd212d73ff42e0b44c9f9"},"schema_version":"1.0"},"canonical_sha256":"1b618ada37f3449c2b344346057d91e1e43a5e53c37c40698ed407aa14d62573","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:41.825486Z","signature_b64":"kLtWR+PwwmQs5w5IcpbGwe9Z/AO5zYxmmtsOA3y8mtSxQmxqHuTbc8EYYGw0loPXORU3EFxQZOxhcrihpvA0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b618ada37f3449c2b344346057d91e1e43a5e53c37c40698ed407aa14d62573","last_reissued_at":"2026-05-17T23:42:41.824729Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:41.824729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.06369","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":"2026-05-17T23:42:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uK+nyP3f35yzq02LwewR1bQxmfZQpCfgW/xW1eOdtrHLigYOyzH6kvuM2aDR5Z7cV2Pk/HQcgimgblYD0cREAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:13:55.149610Z"},"content_sha256":"7e5b2936248d910c8177963c1707260784694d29ce0c4c9156b90e288864124c","schema_version":"1.0","event_id":"sha256:7e5b2936248d910c8177963c1707260784694d29ce0c4c9156b90e288864124c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:DNQYVWRX6NCJYKZUINDAK7MR4H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On triangular numbers, forms of mixed type and their representation numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"B. Ramakrishnan, Lalit Vaishya","submitted_at":"2019-04-12T18:27:09Z","abstract_excerpt":"In \\cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06369","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":"2026-05-17T23:42:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aNrMA8vP1ZMqnR4oSAqACe319ZGn3du24lipf47hPaV1GVRKgvqyRSwek1JwASsaHhxsapPLxgiiOSZReQr1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T19:13:55.149951Z"},"content_sha256":"0ae57a178da066297ee271ebef136eb41c06eeb60c0ded3a81205c24e8843adc","schema_version":"1.0","event_id":"sha256:0ae57a178da066297ee271ebef136eb41c06eeb60c0ded3a81205c24e8843adc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/bundle.json","state_url":"https://pith.science/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/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":"2026-06-25T19:13:55Z","links":{"resolver":"https://pith.science/pith/DNQYVWRX6NCJYKZUINDAK7MR4H","bundle":"https://pith.science/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/bundle.json","state":"https://pith.science/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DNQYVWRX6NCJYKZUINDAK7MR4H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:DNQYVWRX6NCJYKZUINDAK7MR4H","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":"f2cefb4f3981336290b7df8c9e095dd38768bc300befd212d73ff42e0b44c9f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-12T18:27:09Z","title_canon_sha256":"37fd0487741fbb3fe9d145952fab4a40d1aa4c6698cbcc9bfe163de612a851dd"},"schema_version":"1.0","source":{"id":"1904.06369","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06369","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06369v3","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06369","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"pith_short_12","alias_value":"DNQYVWRX6NCJ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"DNQYVWRX6NCJYKZU","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"DNQYVWRX","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:0ae57a178da066297ee271ebef136eb41c06eeb60c0ded3a81205c24e8843adc","target":"graph","created_at":"2026-05-17T23:42:41Z","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":"In \\cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then th","authors_text":"B. Ramakrishnan, Lalit Vaishya","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-12T18:27:09Z","title":"On triangular numbers, forms of mixed type and their representation numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06369","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":"sha256:7e5b2936248d910c8177963c1707260784694d29ce0c4c9156b90e288864124c","target":"record","created_at":"2026-05-17T23:42:41Z","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":"f2cefb4f3981336290b7df8c9e095dd38768bc300befd212d73ff42e0b44c9f9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-12T18:27:09Z","title_canon_sha256":"37fd0487741fbb3fe9d145952fab4a40d1aa4c6698cbcc9bfe163de612a851dd"},"schema_version":"1.0","source":{"id":"1904.06369","kind":"arxiv","version":3}},"canonical_sha256":"1b618ada37f3449c2b344346057d91e1e43a5e53c37c40698ed407aa14d62573","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b618ada37f3449c2b344346057d91e1e43a5e53c37c40698ed407aa14d62573","first_computed_at":"2026-05-17T23:42:41.824729Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:41.824729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kLtWR+PwwmQs5w5IcpbGwe9Z/AO5zYxmmtsOA3y8mtSxQmxqHuTbc8EYYGw0loPXORU3EFxQZOxhcrihpvA0CQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:41.825486Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.06369","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e5b2936248d910c8177963c1707260784694d29ce0c4c9156b90e288864124c","sha256:0ae57a178da066297ee271ebef136eb41c06eeb60c0ded3a81205c24e8843adc"],"state_sha256":"27174978d364eba7d6e748f141ce871a62b364c20795b582055cabaede9b97ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5quIphdR9he8c+zhr1dTKvLeYBJktwLw4vSD7mFLeY5nhX0uWRWP62Pe//zLjFFtLwHLoLSGB8ecBCLSIe6GAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T19:13:55.151866Z","bundle_sha256":"041ea8f437a479d38fe61c1580f4826bfa5012fc5bb80a426231124a3d44e326"}}