{"paper":{"title":"An Efficient Algorithm for Estimating Prime Counts","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artur Samojluk, Artur Siemaszko","submitted_at":"2026-06-30T14:49:55Z","abstract_excerpt":"We propose an efficient algorithm for approximating the prime counting function $\\pi(x)$ using a structured non-uniform partition derived from generalized triangular numbers. The method yields an incremental estimator whose updates require only local computations, resulting in amortized $O(1)$ update complexity and total complexity $O(\\sqrt x)$.\n  A correction term obtained through extensive numerical experimentation significantly improves the approximation accuracy. Computational tests for values up to $10^{19}$\n  show strong agreement with known values of $\\pi(x)$, with accuracy comparable t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31761","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/2606.31761/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"}