{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:GEEXPSHOJVQVCKLHOGOBKF2Y5A","short_pith_number":"pith:GEEXPSHO","schema_version":"1.0","canonical_sha256":"310977c8ee4d61512967719c151758e8384f2f4faefd4f2e8865e3260d62c4a3","source":{"kind":"arxiv","id":"2505.13991","version":1},"attestation_state":"computed","paper":{"title":"The $abc$ conjecture is true almost always","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.NT","authors_text":"Jared Duker Lichtman","submitted_at":"2025-05-20T06:38:56Z","abstract_excerpt":"Let ${\\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\\rm rad}(abc) > K_\\varepsilon\\, c^{1-\\varepsilon}$, for some constant $K_\\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\\in\\{1,\\ldots,N\\}^3$ s"},"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":"2505.13991","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-05-20T06:38:56Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"8de1201a0ea082854b737d1d6cdcf72702c5f747dfe34dbba385e160c6250ce5","abstract_canon_sha256":"c8aefdc1805b2215e06ca15a0d5bb4ecaea75f2c9e5342d3c08e30a834ed82e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:05:54.343111Z","signature_b64":"RGuSHN6a2Db0fFs5tIlwe8rHx93WK00tprpqzpDjuTJx/0/TfQqP5q8hBd1xt47m9p8duYTAJHiK4mp5yFMXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"310977c8ee4d61512967719c151758e8384f2f4faefd4f2e8865e3260d62c4a3","last_reissued_at":"2026-07-05T11:05:54.342693Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:05:54.342693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $abc$ conjecture is true almost always","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.NT","authors_text":"Jared Duker Lichtman","submitted_at":"2025-05-20T06:38:56Z","abstract_excerpt":"Let ${\\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\\rm rad}(abc) > K_\\varepsilon\\, c^{1-\\varepsilon}$, for some constant $K_\\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\\in\\{1,\\ldots,N\\}^3$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.13991","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.13991/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2505.13991","created_at":"2026-07-05T11:05:54.342745+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.13991v1","created_at":"2026-07-05T11:05:54.342745+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.13991","created_at":"2026-07-05T11:05:54.342745+00:00"},{"alias_kind":"pith_short_12","alias_value":"GEEXPSHOJVQV","created_at":"2026-07-05T11:05:54.342745+00:00"},{"alias_kind":"pith_short_16","alias_value":"GEEXPSHOJVQVCKLH","created_at":"2026-07-05T11:05:54.342745+00:00"},{"alias_kind":"pith_short_8","alias_value":"GEEXPSHO","created_at":"2026-07-05T11:05:54.342745+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.02885","citing_title":"On the exceptional set in the $abc$ conjecture","ref_index":2,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A","json":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A.json","graph_json":"https://pith.science/api/pith-number/GEEXPSHOJVQVCKLHOGOBKF2Y5A/graph.json","events_json":"https://pith.science/api/pith-number/GEEXPSHOJVQVCKLHOGOBKF2Y5A/events.json","paper":"https://pith.science/paper/GEEXPSHO"},"agent_actions":{"view_html":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A","download_json":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A.json","view_paper":"https://pith.science/paper/GEEXPSHO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.13991&json=true","fetch_graph":"https://pith.science/api/pith-number/GEEXPSHOJVQVCKLHOGOBKF2Y5A/graph.json","fetch_events":"https://pith.science/api/pith-number/GEEXPSHOJVQVCKLHOGOBKF2Y5A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A/action/storage_attestation","attest_author":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A/action/author_attestation","sign_citation":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A/action/citation_signature","submit_replication":"https://pith.science/pith/GEEXPSHOJVQVCKLHOGOBKF2Y5A/action/replication_record"}},"created_at":"2026-07-05T11:05:54.342745+00:00","updated_at":"2026-07-05T11:05:54.342745+00:00"}