{"paper":{"title":"Solution to the Erdos problem on distinct residues of factorials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"There is no prime number p > 5 such that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct.","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Vyacheslav M. Abramov","submitted_at":"2026-04-29T08:39:51Z","abstract_excerpt":"Paul Erdos posed the following question: Is there a prime number $p>5$ such that the residues of $2!$, $3!$,\\ldots, $(p-1)!$ modulo $p$ all are distinct. In this short note, we give the negative answer on this question in an elementary way."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"There is no prime number p>5 such that the residues of 2!, 3!,…, (p-1)! modulo p all are distinct.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The elementary proof applies to every prime p>5 with no exceptions arising from special cases or additional modular constraints.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"No prime p>5 exists such that the residues of 2!, 3!, ..., (p-1)! modulo p are all distinct.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There is no prime number p > 5 such that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"796e1428ae53c121bcb33bde6ccbe306a425246485f6b924399785538ff1a2bd"},"source":{"id":"2604.26429","kind":"arxiv","version":3},"verdict":{"id":"48459c9c-a3dd-4eaf-8eef-e251cfcf4f44","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T03:17:37.183518Z","strongest_claim":"There is no prime number p>5 such that the residues of 2!, 3!,…, (p-1)! modulo p all are distinct.","one_line_summary":"No prime p>5 exists such that the residues of 2!, 3!, ..., (p-1)! modulo p are all distinct.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The elementary proof applies to every prime p>5 with no exceptions arising from special cases or additional modular constraints.","pith_extraction_headline":"There is no prime number p > 5 such that the residues of 2!, 3!, …, (p-1)! modulo p are all distinct."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26429/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T00:37:01.465827Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:11:20.202705Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"23ce1c029e3d2cd05ce7de3cd4ff7f682c6efed4952b21a1df21647df2a8a89d"},"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"}