{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:5AU5MFWDYKELKBFHWTKKSYNRDV","short_pith_number":"pith:5AU5MFWD","schema_version":"1.0","canonical_sha256":"e829d616c3c288b504a7b4d4a961b11d4aca8fe0462eb129c63cbf65be02f113","source":{"kind":"arxiv","id":"1811.06831","version":1},"attestation_state":"computed","paper":{"title":"The diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $S^{d-2}\\subset S^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mostafa W. Hassan, Naser T. Sardari, Rodrigo Smith, Xiaohan Zhu, Yuchen Mao","submitted_at":"2018-11-12T00:21:33Z","abstract_excerpt":"Assume a polynomial-time algorithm for factoring integers, Conjecture~\\ref{conj}, $d\\geq 3,$ and $q$ and $p$ are prime numbers, where $p\\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\\log(q)$ that lifts every $\\mathbb{Z}/q\\mathbb{Z}$ point of $S^{d-2}\\subset S^{d}$ to a $\\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \\text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $"},"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":"1811.06831","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-11-12T00:21:33Z","cross_cats_sorted":[],"title_canon_sha256":"4a52b84b1b8015c1396ee181e4c531e6e08e44aee55488593b93b881e846ea2d","abstract_canon_sha256":"97ed8f8ba6b07805dcb6b0a9dc13f02dedf35be7a967d7797e962c25a66fd210"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:33.768572Z","signature_b64":"NNKXK14XKIyNKgFNdaNRo2SaJYUQ+NMbWGp7w7S1yp6+O/8DkPwrD+XBt+oc98Byryrw85v9cA59L4eQYWttDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e829d616c3c288b504a7b4d4a961b11d4aca8fe0462eb129c63cbf65be02f113","last_reissued_at":"2026-05-18T00:00:33.768063Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:33.768063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $S^{d-2}\\subset S^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mostafa W. Hassan, Naser T. Sardari, Rodrigo Smith, Xiaohan Zhu, Yuchen Mao","submitted_at":"2018-11-12T00:21:33Z","abstract_excerpt":"Assume a polynomial-time algorithm for factoring integers, Conjecture~\\ref{conj}, $d\\geq 3,$ and $q$ and $p$ are prime numbers, where $p\\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\\log(q)$ that lifts every $\\mathbb{Z}/q\\mathbb{Z}$ point of $S^{d-2}\\subset S^{d}$ to a $\\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \\text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06831","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":"1811.06831","created_at":"2026-05-18T00:00:33.768138+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.06831v1","created_at":"2026-05-18T00:00:33.768138+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.06831","created_at":"2026-05-18T00:00:33.768138+00:00"},{"alias_kind":"pith_short_12","alias_value":"5AU5MFWDYKEL","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"5AU5MFWDYKELKBFH","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"5AU5MFWD","created_at":"2026-05-18T12:32:08.215937+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/5AU5MFWDYKELKBFHWTKKSYNRDV","json":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV.json","graph_json":"https://pith.science/api/pith-number/5AU5MFWDYKELKBFHWTKKSYNRDV/graph.json","events_json":"https://pith.science/api/pith-number/5AU5MFWDYKELKBFHWTKKSYNRDV/events.json","paper":"https://pith.science/paper/5AU5MFWD"},"agent_actions":{"view_html":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV","download_json":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV.json","view_paper":"https://pith.science/paper/5AU5MFWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.06831&json=true","fetch_graph":"https://pith.science/api/pith-number/5AU5MFWDYKELKBFHWTKKSYNRDV/graph.json","fetch_events":"https://pith.science/api/pith-number/5AU5MFWDYKELKBFHWTKKSYNRDV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV/action/storage_attestation","attest_author":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV/action/author_attestation","sign_citation":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV/action/citation_signature","submit_replication":"https://pith.science/pith/5AU5MFWDYKELKBFHWTKKSYNRDV/action/replication_record"}},"created_at":"2026-05-18T00:00:33.768138+00:00","updated_at":"2026-05-18T00:00:33.768138+00:00"}