{"paper":{"title":"Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander R Povolotsky","submitted_at":"2026-05-07T20:41:46Z","abstract_excerpt":"We investigate a ratio sequence derived from the factorization of $p_{m-1} + 1$, where $p_n$ denotes the $n$th prime. For each $m \\geq 3$, write $p_{m-1} + 1 = L_m R_m$ with $L_m$ the largest prime factor. Restricting to those $m$ for which $L_m > m$ (equivalently, $m \\in \\text{A223881}$), we obtain a multiset of values $R_m$. Since $p_{m-1}+1$ is even and $L_m > 3$ is odd, all values of $R_m$ are strictly even. Sorting the distinct $R_m$ by decreasing frequency yields a new sequence beginning $2, 6, 4, 8, 10, 12, 14, 16 \\dots$. This article explains how this construction arises naturally from"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We propose a heuristic asymptotic model explaining the observed frequency ordering via classical results on primes in arithmetic progressions and support the model with numerical log-log analysis.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the observed frequencies of the ratio families R_m are asymptotically governed by the distribution of primes in arithmetic progressions in a manner that can be captured by a heuristic model without post-hoc parameter tuning to the specific data set.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Frequency-ordering the smaller factors R_m in p_{m-1}+1 factorizations for m where the largest factor exceeds m produces a new sequence explained by a heuristic model using the distribution of primes in arithmetic progressions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d0cf63113ea4e52749a3cee69b5ad55a41b7aac2535308bcc20cc91ce7825759"},"source":{"id":"2605.08256","kind":"arxiv","version":2},"verdict":{"id":"0e0550cd-f2ad-411f-a3dc-a8ff87b9aba5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T01:35:55.043285Z","strongest_claim":"We propose a heuristic asymptotic model explaining the observed frequency ordering via classical results on primes in arithmetic progressions and support the model with numerical log-log analysis.","one_line_summary":"Frequency-ordering the smaller factors R_m in p_{m-1}+1 factorizations for m where the largest factor exceeds m produces a new sequence explained by a heuristic model using the distribution of primes in arithmetic progressions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the observed frequencies of the ratio families R_m are asymptotically governed by the distribution of primes in arithmetic progressions in a manner that can be captured by a heuristic model without post-hoc parameter tuning to the specific data set.","pith_extraction_headline":"Frequencies of ratios from factoring p_{m-1}+1 for m in A223881 follow an asymptotic ordering predicted by primes in arithmetic progressions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.08256/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T11:42:03.517307Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T06:39:39.324483Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:19.321630Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T12:14:15.477231Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e239a683cc9beb2a6f6d923dfb0a2411661631e8f01d12d601310447d019ef1d"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"e67fae401968cdf379e180097f805caacc434f12bf8f2f5ebed08a874ddd63a3"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}