{"paper":{"title":"A Ceiling Continued Fraction Approach to the Erd\\H{o}s-Straus Conjecture: Heuristic finiteness of counterexamples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A ceiling continued fraction approach provides heuristic evidence that the Erdős-Straus conjecture has only finitely many counterexamples.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andres Ventas","submitted_at":"2026-05-06T06:56:14Z","abstract_excerpt":"We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than congruence-based techniques. We derive a super-polynomial upper bound on the failure probability; its convergence, together with the Borel-Cantelli lemma, provides heuristic evidence that counterexamples, if any exist, form a finite set. Computational tests on 10^9 primes in ranges around 10^17, 10^52, and 10^131, show no counterexamples with very small search d"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive a super-polynomial upper bound on the failure probability; its convergence, together with the Borel-Cantelli lemma, provides heuristic evidence that counterexamples, if any exist, form a finite set.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the super-polynomial upper bound on failure probability derived from the FCT framework is tight enough and that the failure events across primes satisfy the conditions needed for the Borel-Cantelli lemma to conclude finiteness.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new ceiling continued fraction method finds no counterexamples in searches over 10^9 primes near 10^17 and 10^52 plus 10^7 near 10^131, and derives a super-polynomial failure probability bound whose convergence with the Borel-Cantelli lemma heuristically implies only finitely many counterexamples,","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A ceiling continued fraction approach provides heuristic evidence that the Erdős-Straus conjecture has only finitely many counterexamples.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7ff0940faec8ceb13ee2ffdcf0a6d1edee8c977a8cf6e624a806ac1b3534fdce"},"source":{"id":"2605.04551","kind":"arxiv","version":2},"verdict":{"id":"310e7fc7-91f3-4d0a-b923-cb1f8a0f9717","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T15:56:37.095426Z","strongest_claim":"We derive a super-polynomial upper bound on the failure probability; its convergence, together with the Borel-Cantelli lemma, provides heuristic evidence that counterexamples, if any exist, form a finite set.","one_line_summary":"A new ceiling continued fraction method finds no counterexamples in searches over 10^9 primes near 10^17 and 10^52 plus 10^7 near 10^131, and derives a super-polynomial failure probability bound whose convergence with the Borel-Cantelli lemma heuristically implies only finitely many counterexamples,","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the super-polynomial upper bound on failure probability derived from the FCT framework is tight enough and that the failure events across primes satisfy the conditions needed for the Borel-Cantelli lemma to conclude finiteness.","pith_extraction_headline":"A ceiling continued fraction approach provides heuristic evidence that the Erdős-Straus conjecture has only finitely many counterexamples."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04551/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:40:16.025766Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:20.135323Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:20:24.898780Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"8f673fe68a97fcb5314e0c7f4e710ba85f6d2a3b784f692b9980b69a501da77c"},"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"}