{"paper":{"title":"Folkman's theorem and the primes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Folkman's theorem yields two new proofs that there are infinitely many prime numbers.","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"David J. Fern\\'andez-Bret\\'on","submitted_at":"2025-09-15T15:05:56Z","abstract_excerpt":"We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing literature deriving the infinitude of primes from Ramsey-type theorems."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Folkman's theorem (or Hindman's theorem) can be applied to suitably chosen colorings or subsets of the natural numbers in a way that directly forces the set of primes to be infinite.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Two new proofs of the infinitude of primes are derived from Folkman's theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Folkman's theorem yields two new proofs that there are infinitely many prime numbers.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b7fccaa3723d0aede0f01a8d8801a83e9f57c2a5bfb41057d46b23cbc95a0fe4"},"source":{"id":"2509.12025","kind":"arxiv","version":3},"verdict":{"id":"03136fcf-b29e-4be7-9edb-718ffecfc36c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T16:01:22.114939Z","strongest_claim":"We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem.","one_line_summary":"Two new proofs of the infinitude of primes are derived from Folkman's theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Folkman's theorem (or Hindman's theorem) can be applied to suitably chosen colorings or subsets of the natural numbers in a way that directly forces the set of primes to be infinite.","pith_extraction_headline":"Folkman's theorem yields two new proofs that there are infinitely many prime numbers."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.12025/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":1,"snapshot_sha256":"c5d5247154d5a265401a7d5b58128a0b49da0d3e854d9bbe6aaea26a74ce5194"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}