{"paper":{"title":"A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Roy Quintero-Contreras, Yunier Bello-Cruz","submitted_at":"2026-06-30T15:10:21Z","abstract_excerpt":"In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2{,}906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01270","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/2607.01270/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"}