{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:MTRWVO4DM5VPMNEZ72IYCIZ4J7","short_pith_number":"pith:MTRWVO4D","canonical_record":{"source":{"id":"1901.04838","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-05T15:38:52Z","cross_cats_sorted":[],"title_canon_sha256":"75a07ff0eba3b64f6181dd31bca03c6adc916c29192cd26597ccd785c6f98a50","abstract_canon_sha256":"7b96e1bf5295468851271a0b93ad5a3f772b944c7adc83b547e9a1ee7884177f"},"schema_version":"1.0"},"canonical_sha256":"64e36abb83676af63499fe9181233c4ffeaf5c88d4da05e9b46f6b0527d22754","source":{"kind":"arxiv","id":"1901.04838","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.04838","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"arxiv_version","alias_value":"1901.04838v2","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04838","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"pith_short_12","alias_value":"MTRWVO4DM5VP","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"MTRWVO4DM5VPMNEZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"MTRWVO4D","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:MTRWVO4DM5VPMNEZ72IYCIZ4J7","target":"record","payload":{"canonical_record":{"source":{"id":"1901.04838","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-05T15:38:52Z","cross_cats_sorted":[],"title_canon_sha256":"75a07ff0eba3b64f6181dd31bca03c6adc916c29192cd26597ccd785c6f98a50","abstract_canon_sha256":"7b96e1bf5295468851271a0b93ad5a3f772b944c7adc83b547e9a1ee7884177f"},"schema_version":"1.0"},"canonical_sha256":"64e36abb83676af63499fe9181233c4ffeaf5c88d4da05e9b46f6b0527d22754","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:36.943282Z","signature_b64":"jbkivR9MTu1TYWCKx7MGY96k+V+cIcSKSnoNN2IKAg61PJEjdeAdRICLmvRBBUE70A6f5dQ7Dz5viCzuzCPMDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64e36abb83676af63499fe9181233c4ffeaf5c88d4da05e9b46f6b0527d22754","last_reissued_at":"2026-05-17T23:39:36.942685Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:36.942685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.04838","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2El4EtZGA8KLUXBENITI+3MVD9tvfd67y8arZ1dSeS47vISgOLXDyD1kvBHf/5/ZGq/78OA73dqQrI1GDNwGCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:48:32.160622Z"},"content_sha256":"77401c2a69312e1e09a2497458648c70f80e7c74e1c1dfcddbfef69380cce499","schema_version":"1.0","event_id":"sha256:77401c2a69312e1e09a2497458648c70f80e7c74e1c1dfcddbfef69380cce499"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:MTRWVO4DM5VPMNEZ72IYCIZ4J7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the number of pairs of positive integers $\\mathbf{x, y \\leq H}$ such that $\\mathbf{x^2+y^2+1}$, $\\mathbf{x^2+y^2+2}$ are square-free","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"S. I. Dimitrov","submitted_at":"2019-01-05T15:38:52Z","abstract_excerpt":"In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \\leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04838","kind":"arxiv","version":2},"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":"2026-05-17T23:39:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1A69b592/C2Ft/QxcVp5Dd8d5V1W+Bonkh2/6on8XZ9c6iLKfg4rD7mQC2MSs24I851/5AQb2KhG2IDRcBj+AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:48:32.160970Z"},"content_sha256":"7fd0cd157be528691abc6cfa3fbde2648e8fb18dfb90c28c64168080d0196c71","schema_version":"1.0","event_id":"sha256:7fd0cd157be528691abc6cfa3fbde2648e8fb18dfb90c28c64168080d0196c71"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/bundle.json","state_url":"https://pith.science/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/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":"2026-06-28T17:48:32Z","links":{"resolver":"https://pith.science/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7","bundle":"https://pith.science/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/bundle.json","state":"https://pith.science/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTRWVO4DM5VPMNEZ72IYCIZ4J7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MTRWVO4DM5VPMNEZ72IYCIZ4J7","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":"7b96e1bf5295468851271a0b93ad5a3f772b944c7adc83b547e9a1ee7884177f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-05T15:38:52Z","title_canon_sha256":"75a07ff0eba3b64f6181dd31bca03c6adc916c29192cd26597ccd785c6f98a50"},"schema_version":"1.0","source":{"id":"1901.04838","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.04838","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"arxiv_version","alias_value":"1901.04838v2","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04838","created_at":"2026-05-17T23:39:36Z"},{"alias_kind":"pith_short_12","alias_value":"MTRWVO4DM5VP","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"MTRWVO4DM5VPMNEZ","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"MTRWVO4D","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:7fd0cd157be528691abc6cfa3fbde2648e8fb18dfb90c28c64168080d0196c71","target":"graph","created_at":"2026-05-17T23:39:36Z","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 the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \\leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.","authors_text":"S. I. Dimitrov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-05T15:38:52Z","title":"On the number of pairs of positive integers $\\mathbf{x, y \\leq H}$ such that $\\mathbf{x^2+y^2+1}$, $\\mathbf{x^2+y^2+2}$ are square-free"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04838","kind":"arxiv","version":2},"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":"sha256:77401c2a69312e1e09a2497458648c70f80e7c74e1c1dfcddbfef69380cce499","target":"record","created_at":"2026-05-17T23:39:36Z","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":"7b96e1bf5295468851271a0b93ad5a3f772b944c7adc83b547e9a1ee7884177f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-05T15:38:52Z","title_canon_sha256":"75a07ff0eba3b64f6181dd31bca03c6adc916c29192cd26597ccd785c6f98a50"},"schema_version":"1.0","source":{"id":"1901.04838","kind":"arxiv","version":2}},"canonical_sha256":"64e36abb83676af63499fe9181233c4ffeaf5c88d4da05e9b46f6b0527d22754","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64e36abb83676af63499fe9181233c4ffeaf5c88d4da05e9b46f6b0527d22754","first_computed_at":"2026-05-17T23:39:36.942685Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:36.942685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jbkivR9MTu1TYWCKx7MGY96k+V+cIcSKSnoNN2IKAg61PJEjdeAdRICLmvRBBUE70A6f5dQ7Dz5viCzuzCPMDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:36.943282Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.04838","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77401c2a69312e1e09a2497458648c70f80e7c74e1c1dfcddbfef69380cce499","sha256:7fd0cd157be528691abc6cfa3fbde2648e8fb18dfb90c28c64168080d0196c71"],"state_sha256":"1330ba21f3638cdff8b2d357b0817af9b67ae8829c850406753b5803ded7c121"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8nA8zvOjI1wTgkIQ9+Oejhn4ZqdHMgaATonn5akpy+hslzDxkSLe+xsVVcZLGPdsqXBv84tK+jxqEjNgWhiXBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T17:48:32.162815Z","bundle_sha256":"4eddbef1252a3df15730a7aa2b597abe8456ebe5a2de192db0061fcf3e86dbe6"}}