{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:HT3Q4STE7TWK6TYPAW6US66Z7M","short_pith_number":"pith:HT3Q4STE","schema_version":"1.0","canonical_sha256":"3cf70e4a64fcecaf4f0f05bd497bd9fb0917a3a581aa66442e1532281c2a192d","source":{"kind":"arxiv","id":"1906.11128","version":1},"attestation_state":"computed","paper":{"title":"Gaps between divisible terms in $a^2 (a^2 + 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tsz Ho Chan","submitted_at":"2019-06-26T14:36:37Z","abstract_excerpt":"Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \\gg a (\\log a)^{1/8} / (\\log \\log a)^{12}$. We also obtain $b \\gg_\\epsilon a^{15/14 - \\epsilon}$ under the abc conjecture."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1906.11128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-06-26T14:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"6227e9c0502b47a43e7293b51d543133e8f60466dfee6487eb13550c02d78f70","abstract_canon_sha256":"2213a7a135903c8346d512d7fdc7dea5e31aae1155d9dae6d55ac55f2459dacb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:09.749252Z","signature_b64":"iMhUNrNxMSJROFbBmtnSkZwptj6wWCoi0jyxTbXnmvtkov3Gkndsip/8tGx6mLF8FwMOoJNG4G/H7GI7Se4EAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cf70e4a64fcecaf4f0f05bd497bd9fb0917a3a581aa66442e1532281c2a192d","last_reissued_at":"2026-05-17T23:42:09.748789Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:09.748789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaps between divisible terms in $a^2 (a^2 + 1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tsz Ho Chan","submitted_at":"2019-06-26T14:36:37Z","abstract_excerpt":"Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \\gg a (\\log a)^{1/8} / (\\log \\log a)^{12}$. We also obtain $b \\gg_\\epsilon a^{15/14 - \\epsilon}$ under the abc conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.11128","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1906.11128","created_at":"2026-05-17T23:42:09.748855+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.11128v1","created_at":"2026-05-17T23:42:09.748855+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.11128","created_at":"2026-05-17T23:42:09.748855+00:00"},{"alias_kind":"pith_short_12","alias_value":"HT3Q4STE7TWK","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"HT3Q4STE7TWK6TYP","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"HT3Q4STE","created_at":"2026-05-18T12:33:18.533446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M","json":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M.json","graph_json":"https://pith.science/api/pith-number/HT3Q4STE7TWK6TYPAW6US66Z7M/graph.json","events_json":"https://pith.science/api/pith-number/HT3Q4STE7TWK6TYPAW6US66Z7M/events.json","paper":"https://pith.science/paper/HT3Q4STE"},"agent_actions":{"view_html":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M","download_json":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M.json","view_paper":"https://pith.science/paper/HT3Q4STE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.11128&json=true","fetch_graph":"https://pith.science/api/pith-number/HT3Q4STE7TWK6TYPAW6US66Z7M/graph.json","fetch_events":"https://pith.science/api/pith-number/HT3Q4STE7TWK6TYPAW6US66Z7M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M/action/storage_attestation","attest_author":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M/action/author_attestation","sign_citation":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M/action/citation_signature","submit_replication":"https://pith.science/pith/HT3Q4STE7TWK6TYPAW6US66Z7M/action/replication_record"}},"created_at":"2026-05-17T23:42:09.748855+00:00","updated_at":"2026-05-17T23:42:09.748855+00:00"}