{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:RDTWPJT7Y2ZONVDMOLZKAQ7HJH","short_pith_number":"pith:RDTWPJT7","canonical_record":{"source":{"id":"1108.0096","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-30T18:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"29fdc60fb39f0cd1061eabc661da589f416cec49b047af3dc2bcf38fe4f4f4ea","abstract_canon_sha256":"89af986caf74015921deb7a44b52ba2f7c08aa0677ec8a1ae0f39974094204fe"},"schema_version":"1.0"},"canonical_sha256":"88e767a67fc6b2e6d46c72f2a043e749ccd66732c342b52d25bd89076a8603c2","source":{"kind":"arxiv","id":"1108.0096","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0096","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0096v1","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0096","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"pith_short_12","alias_value":"RDTWPJT7Y2ZO","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RDTWPJT7Y2ZONVDM","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RDTWPJT7","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:RDTWPJT7Y2ZONVDMOLZKAQ7HJH","target":"record","payload":{"canonical_record":{"source":{"id":"1108.0096","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-30T18:22:10Z","cross_cats_sorted":[],"title_canon_sha256":"29fdc60fb39f0cd1061eabc661da589f416cec49b047af3dc2bcf38fe4f4f4ea","abstract_canon_sha256":"89af986caf74015921deb7a44b52ba2f7c08aa0677ec8a1ae0f39974094204fe"},"schema_version":"1.0"},"canonical_sha256":"88e767a67fc6b2e6d46c72f2a043e749ccd66732c342b52d25bd89076a8603c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:31.693028Z","signature_b64":"QYFkNDdM9q0MhOqvQc0wRXClR2BH4GNuhD3YqA7v1DlZUz/O3JR78x7V8npw0JeA71JrfrShEX3LgqRdbtnuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88e767a67fc6b2e6d46c72f2a043e749ccd66732c342b52d25bd89076a8603c2","last_reissued_at":"2026-05-18T04:16:31.692613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:31.692613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.0096","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6XfDh9j6yBtUshpmnUw3irl60MVKqN1276nAnNIJKrbKGz1LSw4X9SAWYrOqdsLNcYsat2qlhPCBe+mwOnjxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:08:57.643647Z"},"content_sha256":"14805c3f7d6743fec27620daf55875baa34c9eec344b57c7f6a71965eead6cb8","schema_version":"1.0","event_id":"sha256:14805c3f7d6743fec27620daf55875baa34c9eec344b57c7f6a71965eead6cb8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:RDTWPJT7Y2ZONVDMOLZKAQ7HJH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean Value Theorems for Binary Egyptian Fractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jing-Jing Huang, Robert C. Vaughan","submitted_at":"2011-07-30T18:22:10Z","abstract_excerpt":"In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\\frac{a}{n}=\\frac{1}{x}+\\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0096","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8tGd/M0ekHmbEUDPNex38Z7L5jtAL6A8IvAszZx0jgnQRaMlIBSfabo1c12kdlHAVMXHLY83bErZtz497LilBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:08:57.644054Z"},"content_sha256":"7a460818cc51f9d7261e3e661f7ffc9724966a937b24ff6a06cea32a3b8d5b8e","schema_version":"1.0","event_id":"sha256:7a460818cc51f9d7261e3e661f7ffc9724966a937b24ff6a06cea32a3b8d5b8e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/bundle.json","state_url":"https://pith.science/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/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":"2026-06-04T22:08:57Z","links":{"resolver":"https://pith.science/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH","bundle":"https://pith.science/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/bundle.json","state":"https://pith.science/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RDTWPJT7Y2ZONVDMOLZKAQ7HJH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:RDTWPJT7Y2ZONVDMOLZKAQ7HJH","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":"89af986caf74015921deb7a44b52ba2f7c08aa0677ec8a1ae0f39974094204fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-30T18:22:10Z","title_canon_sha256":"29fdc60fb39f0cd1061eabc661da589f416cec49b047af3dc2bcf38fe4f4f4ea"},"schema_version":"1.0","source":{"id":"1108.0096","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0096","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0096v1","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0096","created_at":"2026-05-18T04:16:31Z"},{"alias_kind":"pith_short_12","alias_value":"RDTWPJT7Y2ZO","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RDTWPJT7Y2ZONVDM","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RDTWPJT7","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:7a460818cc51f9d7261e3e661f7ffc9724966a937b24ff6a06cea32a3b8d5b8e","target":"graph","created_at":"2026-05-18T04:16:31Z","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":"In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\\frac{a}{n}=\\frac{1}{x}+\\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.","authors_text":"Jing-Jing Huang, Robert C. Vaughan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-30T18:22:10Z","title":"Mean Value Theorems for Binary Egyptian Fractions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0096","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":"sha256:14805c3f7d6743fec27620daf55875baa34c9eec344b57c7f6a71965eead6cb8","target":"record","created_at":"2026-05-18T04:16:31Z","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":"89af986caf74015921deb7a44b52ba2f7c08aa0677ec8a1ae0f39974094204fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-30T18:22:10Z","title_canon_sha256":"29fdc60fb39f0cd1061eabc661da589f416cec49b047af3dc2bcf38fe4f4f4ea"},"schema_version":"1.0","source":{"id":"1108.0096","kind":"arxiv","version":1}},"canonical_sha256":"88e767a67fc6b2e6d46c72f2a043e749ccd66732c342b52d25bd89076a8603c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88e767a67fc6b2e6d46c72f2a043e749ccd66732c342b52d25bd89076a8603c2","first_computed_at":"2026-05-18T04:16:31.692613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:31.692613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QYFkNDdM9q0MhOqvQc0wRXClR2BH4GNuhD3YqA7v1DlZUz/O3JR78x7V8npw0JeA71JrfrShEX3LgqRdbtnuDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:31.693028Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0096","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14805c3f7d6743fec27620daf55875baa34c9eec344b57c7f6a71965eead6cb8","sha256:7a460818cc51f9d7261e3e661f7ffc9724966a937b24ff6a06cea32a3b8d5b8e"],"state_sha256":"4ce4b4206df139f703765393e64f277d0b970fe50987b67696a1f88faf84f74b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/LSWCudDq7W/FghZfWHLCbkEpxrSstQhSrH9izhrBEi8tlUWvSQpfAenD7b8NKhPoD56zaxpcJEf5imWpntVDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T22:08:57.646523Z","bundle_sha256":"3f11f9ec49e63631f4b2fba81264761fff4d65c0bface9f91f027697119d5076"}}