{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:G2C347S7JOOASIVGJFPTE2QFJI","short_pith_number":"pith:G2C347S7","canonical_record":{"source":{"id":"1007.1526","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","cross_cats_sorted":[],"title_canon_sha256":"fa0e26ac9d2d3cb0a4b3ac8e9b2ddd258bf89352fb01241a0b8573a0a7421063","abstract_canon_sha256":"cf46b2b35b5a2cb3b4a7b67c12df33d588031b845edd016c256c08b5df9e39b8"},"schema_version":"1.0"},"canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","source":{"kind":"arxiv","id":"1007.1526","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1526","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1526v2","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1526","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"pith_short_12","alias_value":"G2C347S7JOOA","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G2C347S7JOOASIVG","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G2C347S7","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:G2C347S7JOOASIVGJFPTE2QFJI","target":"record","payload":{"canonical_record":{"source":{"id":"1007.1526","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","cross_cats_sorted":[],"title_canon_sha256":"fa0e26ac9d2d3cb0a4b3ac8e9b2ddd258bf89352fb01241a0b8573a0a7421063","abstract_canon_sha256":"cf46b2b35b5a2cb3b4a7b67c12df33d588031b845edd016c256c08b5df9e39b8"},"schema_version":"1.0"},"canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:27.640849Z","signature_b64":"LyJIEXP3NuJtZcsNsxl1bQaG1b9bfoaDjcRZYCi1P2URM9wC1dVsVw8R1CyeIuwK8jPi61BfbmRccavBxnqXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","last_reissued_at":"2026-05-18T04:39:27.640409Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:27.640409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.1526","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"14/zvuo/t2ozByfydlkrTVH3si7DWfmBNQHmsmdMUYPy2Va20ukqNgdKgHZkSRSBqexvaBk1ONXAF9Gbj9RwBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T13:17:58.690159Z"},"content_sha256":"e2f967622d2f14a3d1868950458ef11aa4223bd0dfb4e6fdffe9010d19ac04d0","schema_version":"1.0","event_id":"sha256:e2f967622d2f14a3d1868950458ef11aa4223bd0dfb4e6fdffe9010d19ac04d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:G2C347S7JOOASIVGJFPTE2QFJI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concentration points on two and three dimensional modular hyperbolas and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J. Cilleruelo, M. Z. Garaev","submitted_at":"2010-07-09T08:04:03Z","abstract_excerpt":"Let $p$ be a large prime number, $K,L,M,\\lambda$ be integers with $1\\le M\\le p$ and ${\\color{red}\\gcd}(\\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\\equiv\\lambda \\pmod p, \\qquad K+1\\le x\\le K+M,\\quad L+1\\le y\\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\\equiv\\lambda\\pmod p, \\quad L+1\\le x,y,z\\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\\le M^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1526","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zh+44t60spQkw2AT2BPnrNTAQJdMumwEJZgsma4s33WGhG9jtjrNnBFPDGqvOS5P4BMLrnZhkSzA9HXyPRG3AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T13:17:58.690515Z"},"content_sha256":"105f048a6994bf0745df271a88430e9cf0ba7e2a14141aeafba668b9d855478b","schema_version":"1.0","event_id":"sha256:105f048a6994bf0745df271a88430e9cf0ba7e2a14141aeafba668b9d855478b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/bundle.json","state_url":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G2C347S7JOOASIVGJFPTE2QFJI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T13:17:58Z","links":{"resolver":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI","bundle":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/bundle.json","state":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G2C347S7JOOASIVGJFPTE2QFJI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:G2C347S7JOOASIVGJFPTE2QFJI","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":"cf46b2b35b5a2cb3b4a7b67c12df33d588031b845edd016c256c08b5df9e39b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","title_canon_sha256":"fa0e26ac9d2d3cb0a4b3ac8e9b2ddd258bf89352fb01241a0b8573a0a7421063"},"schema_version":"1.0","source":{"id":"1007.1526","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1526","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1526v2","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1526","created_at":"2026-05-18T04:39:27Z"},{"alias_kind":"pith_short_12","alias_value":"G2C347S7JOOA","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G2C347S7JOOASIVG","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G2C347S7","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:105f048a6994bf0745df271a88430e9cf0ba7e2a14141aeafba668b9d855478b","target":"graph","created_at":"2026-05-18T04:39:27Z","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":"Let $p$ be a large prime number, $K,L,M,\\lambda$ be integers with $1\\le M\\le p$ and ${\\color{red}\\gcd}(\\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\\equiv\\lambda \\pmod p, \\qquad K+1\\le x\\le K+M,\\quad L+1\\le y\\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\\equiv\\lambda\\pmod p, \\quad L+1\\le x,y,z\\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\\le M^{","authors_text":"J. Cilleruelo, M. Z. Garaev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","title":"Concentration points on two and three dimensional modular hyperbolas and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1526","kind":"arxiv","version":2},"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:e2f967622d2f14a3d1868950458ef11aa4223bd0dfb4e6fdffe9010d19ac04d0","target":"record","created_at":"2026-05-18T04:39:27Z","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":"cf46b2b35b5a2cb3b4a7b67c12df33d588031b845edd016c256c08b5df9e39b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","title_canon_sha256":"fa0e26ac9d2d3cb0a4b3ac8e9b2ddd258bf89352fb01241a0b8573a0a7421063"},"schema_version":"1.0","source":{"id":"1007.1526","kind":"arxiv","version":2}},"canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","first_computed_at":"2026-05-18T04:39:27.640409Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:27.640409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LyJIEXP3NuJtZcsNsxl1bQaG1b9bfoaDjcRZYCi1P2URM9wC1dVsVw8R1CyeIuwK8jPi61BfbmRccavBxnqXBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:27.640849Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.1526","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e2f967622d2f14a3d1868950458ef11aa4223bd0dfb4e6fdffe9010d19ac04d0","sha256:105f048a6994bf0745df271a88430e9cf0ba7e2a14141aeafba668b9d855478b"],"state_sha256":"3a822fd29633d92472ce1894d3f2c18142e58a9523cd0d7a4c250981ef9e5cfd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WYcvV3w8Je0paowRSHH+Y0OiZw9MZsHIB8zSu9Ul4bEqHNziNCoJO/3/+ZQUU+ED6TG4uN9kw6OQyka6NTsZDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T13:17:58.692369Z","bundle_sha256":"14ea619ce3830f47f1ea40abf42434c83e51d59b42e243d29e1ca087f6bdef23"}}