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By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\\Z_p \\times \\Z_{p^2}) = 2p$ and $T(\\Z_p \\times \\Z_{pq}) = p+1$. Via Gr\\\"obner bases, we compute $T(\\Z_m \\times \\Z_n)$ for $2 \\leq m \\leq 7$ and $2 \\leq n \\leq 19$."},"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":"1203.6604","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-29T17:41:02Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"2e4bcfb870c65733e908045c62030f61577660f264a64c7092e48588e89d1af5","abstract_canon_sha256":"1b64f95b2ff46979e31565ac9a05f0e775934cc9c6cf6e71b22e198c479e7261"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:02.072182Z","signature_b64":"0WaLXwjOepIJ0vbWkVZ08e271iCx4WwRtiSY+WYhxR81ol+Fh84AqUamf1anxnIwlGaLXT6r/et8zFR+oHmtDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aec306631ef800e2f288005477d471a997d85c998811254249369ab80df2e55e","last_reissued_at":"2026-05-18T03:59:02.071368Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:02.071368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The no-three-in-line problem on a torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Andrew Groot, Bart Snapp, Deven Pandya, Jim Fowler","submitted_at":"2012-03-29T17:41:02Z","abstract_excerpt":"Let $T(\\Z_m \\times \\Z_n)$ denote the maximal number of points that can be placed on an $m \\times n$ discrete torus with \"no three in a line,\" meaning no three in a coset of a cyclic subgroup of $\\Z_m \\times \\Z_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\\Z_p \\times \\Z_{p^2}) = 2p$ and $T(\\Z_p \\times \\Z_{pq}) = p+1$. Via Gr\\\"obner bases, we compute $T(\\Z_m \\times \\Z_n)$ for $2 \\leq m \\leq 7$ and $2 \\leq n \\leq 19$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6604","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":"1203.6604","created_at":"2026-05-18T03:59:02.071492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.6604v1","created_at":"2026-05-18T03:59:02.071492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6604","created_at":"2026-05-18T03:59:02.071492+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3BQMYY67AAO","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3BQMYY67AAOF4UI","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3BQMYY6","created_at":"2026-05-18T12:27:25.539911+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.09215","citing_title":"No-three-in-line sets on the checkerboard grid","ref_index":18,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG","json":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG.json","graph_json":"https://pith.science/api/pith-number/V3BQMYY67AAOF4UIABKHPVDRVG/graph.json","events_json":"https://pith.science/api/pith-number/V3BQMYY67AAOF4UIABKHPVDRVG/events.json","paper":"https://pith.science/paper/V3BQMYY6"},"agent_actions":{"view_html":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG","download_json":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG.json","view_paper":"https://pith.science/paper/V3BQMYY6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.6604&json=true","fetch_graph":"https://pith.science/api/pith-number/V3BQMYY67AAOF4UIABKHPVDRVG/graph.json","fetch_events":"https://pith.science/api/pith-number/V3BQMYY67AAOF4UIABKHPVDRVG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG/action/storage_attestation","attest_author":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG/action/author_attestation","sign_citation":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG/action/citation_signature","submit_replication":"https://pith.science/pith/V3BQMYY67AAOF4UIABKHPVDRVG/action/replication_record"}},"created_at":"2026-05-18T03:59:02.071492+00:00","updated_at":"2026-05-18T03:59:02.071492+00:00"}