{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WRL276IHCYKCTMX4G4IS5SOOWW","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":"a09b1effe8e948293285d76eb13089fedb109a3f46bfdb02e18ff55e20dca766","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T11:07:51Z","title_canon_sha256":"12e224a285d23d18ed4c66f86740b687ac53b6cc8a90510b6707964e43e81cba"},"schema_version":"1.0","source":{"id":"1805.02945","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.02945","created_at":"2026-05-18T00:16:35Z"},{"alias_kind":"arxiv_version","alias_value":"1805.02945v1","created_at":"2026-05-18T00:16:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.02945","created_at":"2026-05-18T00:16:35Z"},{"alias_kind":"pith_short_12","alias_value":"WRL276IHCYKC","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WRL276IHCYKCTMX4","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WRL276IH","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:7c00178b8262486f0ba074d4ea0252640e292da342620cfae43786731ae7b0b0","target":"graph","created_at":"2026-05-18T00:16:35Z","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 prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\\mathcal{O}_{\\epsilon}(n^{3/5+\\epsilon})$ solutions of $\\frac{m}{n}=\\frac{1}{a_1}+\\frac{1}{a_2}+\\frac{1}{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when $m=4$ and $n$ is a prime. Moreover there exists an algorithm finding all solutions in expected running time $\\mathcal{O}_{\\epsilon}\\left(n^{\\epsilon}\\left(\\frac{n^3}{m^2}\\right)^{1/5}\\r","authors_text":"Christian Elsholtz, Stefan Planitzer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T11:07:51Z","title":"The number of solutions of the Erd\\H{o}s-Straus Equation and sums of $k$ unit fractions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.02945","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":"sha256:523fa27009d3b5136ad208795ce877ba71da92c39ab958cd953e774586fca02d","target":"record","created_at":"2026-05-18T00:16:35Z","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":"a09b1effe8e948293285d76eb13089fedb109a3f46bfdb02e18ff55e20dca766","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-05-08T11:07:51Z","title_canon_sha256":"12e224a285d23d18ed4c66f86740b687ac53b6cc8a90510b6707964e43e81cba"},"schema_version":"1.0","source":{"id":"1805.02945","kind":"arxiv","version":1}},"canonical_sha256":"b457aff907161429b2fc37112ec9ceb591d34b94babd1ef1f11322467619edb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b457aff907161429b2fc37112ec9ceb591d34b94babd1ef1f11322467619edb3","first_computed_at":"2026-05-18T00:16:35.164596Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:35.164596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C8r6SdtLHzWt9NRkhP787Y/zlWqIagwNeEzx/9sa9cpRfXqc6vXGTDK9BMdNy0zngyrTcn5ffzS72djYs07eDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:35.165248Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.02945","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:523fa27009d3b5136ad208795ce877ba71da92c39ab958cd953e774586fca02d","sha256:7c00178b8262486f0ba074d4ea0252640e292da342620cfae43786731ae7b0b0"],"state_sha256":"a36b694539a814b9363f6bcfcbec749f6e20d5c5dfb37b5d231cd04ea4648950"}