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In this article, we consider such a set of rational points restricted to a given hyperbola.\n  To be precise, for rational numbers $a$, $b$, $c$, and $d$ such that the quantity $D = \\bigl(a \\, d - b \\, c \\bigr) / \\bigl(2 \\, a^2 \\bigr)$ is defined and nonzero, we consider rational distance sets on the conic section $a \\, x \\, y + b \\, x + c \\, y + d = 0$. We show that, if the elliptic curve $Y^2 = X^3 - D^2 \\, X$ has infinitely many rational points, then there a"},"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":"1108.0690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-02T20:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"6266c07394068a37d9ab7c4dc697068ab86d8eb4fceb5e9f70a505c9eb178260","abstract_canon_sha256":"6a5460e1fc2571a6baaa79a7e8b077c2b09ba2937a7a5acdee8357418b93a9c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:22.148199Z","signature_b64":"MR3j/RWrscIry3YVcmHgjzOKgX+nwlf/4PIHWwoAGP/YT8oE0SjbRuxW9dLDpHoFd5iJTMoPOSoQQlBReo0XAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c96aa86476a62d8d312e6f2b5a422212c5133c02a636e7aa5429589c20d7b1fa","last_reissued_at":"2026-05-18T04:16:22.147693Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:22.147693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Points on Hyperbolas at Rational Distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Edray Herber Goins, Kevin Mugo","submitted_at":"2011-08-02T20:52:22Z","abstract_excerpt":"Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. 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We show that, if the elliptic curve $Y^2 = X^3 - D^2 \\, X$ has infinitely many rational points, then there a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0690","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":"1108.0690","created_at":"2026-05-18T04:16:22.147769+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.0690v1","created_at":"2026-05-18T04:16:22.147769+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0690","created_at":"2026-05-18T04:16:22.147769+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZFVKQZDWUYWY","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZFVKQZDWUYWY2MJO","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZFVKQZDW","created_at":"2026-05-18T12:26:47.523578+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/ZFVKQZDWUYWY2MJON4VVUQRCCL","json":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL.json","graph_json":"https://pith.science/api/pith-number/ZFVKQZDWUYWY2MJON4VVUQRCCL/graph.json","events_json":"https://pith.science/api/pith-number/ZFVKQZDWUYWY2MJON4VVUQRCCL/events.json","paper":"https://pith.science/paper/ZFVKQZDW"},"agent_actions":{"view_html":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL","download_json":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL.json","view_paper":"https://pith.science/paper/ZFVKQZDW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.0690&json=true","fetch_graph":"https://pith.science/api/pith-number/ZFVKQZDWUYWY2MJON4VVUQRCCL/graph.json","fetch_events":"https://pith.science/api/pith-number/ZFVKQZDWUYWY2MJON4VVUQRCCL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL/action/storage_attestation","attest_author":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL/action/author_attestation","sign_citation":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL/action/citation_signature","submit_replication":"https://pith.science/pith/ZFVKQZDWUYWY2MJON4VVUQRCCL/action/replication_record"}},"created_at":"2026-05-18T04:16:22.147769+00:00","updated_at":"2026-05-18T04:16:22.147769+00:00"}