{"paper":{"title":"A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Wujie Shi","submitted_at":"2026-05-19T17:23:39Z","abstract_excerpt":"Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \\geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \\emph{matrix multiplicity} $r(n)$, the number of times $n$ appears in $B$. We prove the exact identity \\[ P_k = N_k - S_k + E_k \\] where $P_k = \\#\\{\\text{primes in } I_k\\}$, $N_k = 4k$ counts the odd integers in $I_k$, $S_k = \\sum_{n \\in I_k \\text{ odd}} r(n)$ is the total matrix multiplicity, and $E_k = \\sum_{n \\in I_k \\text{ odd}} (r(n)-1)$ measures the excess multiplicity of non-semiprime odd "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21529","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/2605.21529/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"}