{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:URHMYYDW4NV7ZEMTPRRUISX5FA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9b7208f255b9a86f5660df307613a773934b8665f3788000f3024b651a8e5e2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-20T02:34:24Z","title_canon_sha256":"2d7e7c462a59f8ee6a891a0f7d5eb7e2a9bf1d811aff89905fd850cf5a7ced5a"},"schema_version":"1.0","source":{"id":"1112.4547","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.4547","created_at":"2026-05-18T04:05:58Z"},{"alias_kind":"arxiv_version","alias_value":"1112.4547v1","created_at":"2026-05-18T04:05:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4547","created_at":"2026-05-18T04:05:58Z"},{"alias_kind":"pith_short_12","alias_value":"URHMYYDW4NV7","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"URHMYYDW4NV7ZEMT","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"URHMYYDW","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:7074aea5ff851dbcaad22609820fe957cf4ab74383e6c371386b783acfc04145","target":"graph","created_at":"2026-05-18T04:05:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $s>0$. Previous work showed that there are nine essentially distinct $(a,b,c,r,s)$ for which $N \\ge 4$, except possibly for cases in which the solutions have $r$, $a$, $x$, $s$, $b$, and $y$ each bounded by $8 \\cdot 10^{14}$ or $2 \\cdot 10^{15}$. In this paper we show that there are no further cases with $N \\ge 4$ within these bounds. We note that $N = 3$ for an infinite number ","authors_text":"Reese Scott, Robert Styer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-20T02:34:24Z","title":"Handling a large bound for a problem on the generalized Pillai equation $\\pm r a^x \\pm s b^y = c$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4547","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d81d99301a15ec988287278e22036eab76c6cc1917efd4425c6ee38b4489da9","target":"record","created_at":"2026-05-18T04:05:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9b7208f255b9a86f5660df307613a773934b8665f3788000f3024b651a8e5e2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-20T02:34:24Z","title_canon_sha256":"2d7e7c462a59f8ee6a891a0f7d5eb7e2a9bf1d811aff89905fd850cf5a7ced5a"},"schema_version":"1.0","source":{"id":"1112.4547","kind":"arxiv","version":1}},"canonical_sha256":"a44ecc6076e36bfc91937c63444afd2815e296fda6a38b4db6d85354294ee785","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a44ecc6076e36bfc91937c63444afd2815e296fda6a38b4db6d85354294ee785","first_computed_at":"2026-05-18T04:05:58.603924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:58.603924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q8rZmp6UKeSC9MOK6rfLJLiOKkc2p86aSCZMmwwajYBwGz6GPWoLasl/WmKKO4H3GDkrUPZqOZ/xltJQLYhdCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:58.604555Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.4547","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d81d99301a15ec988287278e22036eab76c6cc1917efd4425c6ee38b4489da9","sha256:7074aea5ff851dbcaad22609820fe957cf4ab74383e6c371386b783acfc04145"],"state_sha256":"384b6efec5a70d9743552e912d0ba1aad543b21fbf2b7d601cf749a72d1030c2"}