{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:UPXYB7C7X7MSXJXXPUF25ANAR5","short_pith_number":"pith:UPXYB7C7","schema_version":"1.0","canonical_sha256":"a3ef80fc5fbfd92ba6f77d0bae81a08f705afcaa0c3ed2a75d285ee6985a283d","source":{"kind":"arxiv","id":"1102.4833","version":1},"attestation_state":"computed","paper":{"title":"The generalized Pillai equation $\\pm r a^x \\pm s b^y = c$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Reese Scott, Robert Styer","submitted_at":"2011-02-22T03:18:42Z","abstract_excerpt":"We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \\in \\{0,1\\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When $(ra,sb)=1$, we show that $N \\le 3$ except for a finite number of cases all of which satisfy $\\max(a,b,r,s, x,y) < 2 \\cdot 10^{15}$ for each solution; when $(a,b)>1$, we show that $N \\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of infinite families of cases giving N=3 solutions."},"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":"1102.4833","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-22T03:18:42Z","cross_cats_sorted":[],"title_canon_sha256":"7bcaa10f13e40212256f708d74d91e9096d16b2435fd6fc4fad253201b59d663","abstract_canon_sha256":"b727a15687193752e9212859a0b79e0102ded7188e4fc9778a699b931438727d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:07.254422Z","signature_b64":"Zia1qbz9t5dZ4aGjvLHDFZKq5ZpaYdCJ9uRErILY5yDWoQKXNNzfhSmH97KDxs/nS2ZmVR1c5S4TSX4ZW4BkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3ef80fc5fbfd92ba6f77d0bae81a08f705afcaa0c3ed2a75d285ee6985a283d","last_reissued_at":"2026-05-18T04:28:07.253836Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:07.253836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The generalized Pillai equation $\\pm r a^x \\pm s b^y = c$, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Reese Scott, Robert Styer","submitted_at":"2011-02-22T03:18:42Z","abstract_excerpt":"We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \\in \\{0,1\\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When $(ra,sb)=1$, we show that $N \\le 3$ except for a finite number of cases all of which satisfy $\\max(a,b,r,s, x,y) < 2 \\cdot 10^{15}$ for each solution; when $(a,b)>1$, we show that $N \\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of infinite families of cases giving N=3 solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4833","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":"1102.4833","created_at":"2026-05-18T04:28:07.253928+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4833v1","created_at":"2026-05-18T04:28:07.253928+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4833","created_at":"2026-05-18T04:28:07.253928+00:00"},{"alias_kind":"pith_short_12","alias_value":"UPXYB7C7X7MS","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"UPXYB7C7X7MSXJXX","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"UPXYB7C7","created_at":"2026-05-18T12:26:42.757692+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/UPXYB7C7X7MSXJXXPUF25ANAR5","json":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5.json","graph_json":"https://pith.science/api/pith-number/UPXYB7C7X7MSXJXXPUF25ANAR5/graph.json","events_json":"https://pith.science/api/pith-number/UPXYB7C7X7MSXJXXPUF25ANAR5/events.json","paper":"https://pith.science/paper/UPXYB7C7"},"agent_actions":{"view_html":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5","download_json":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5.json","view_paper":"https://pith.science/paper/UPXYB7C7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4833&json=true","fetch_graph":"https://pith.science/api/pith-number/UPXYB7C7X7MSXJXXPUF25ANAR5/graph.json","fetch_events":"https://pith.science/api/pith-number/UPXYB7C7X7MSXJXXPUF25ANAR5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5/action/storage_attestation","attest_author":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5/action/author_attestation","sign_citation":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5/action/citation_signature","submit_replication":"https://pith.science/pith/UPXYB7C7X7MSXJXXPUF25ANAR5/action/replication_record"}},"created_at":"2026-05-18T04:28:07.253928+00:00","updated_at":"2026-05-18T04:28:07.253928+00:00"}