{"paper":{"title":"Positive density for Sun's $2^k+m$ conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Natural numbers n that can be written as n = k + m with 2^k + m prime have positive asymptotic density at least 0.0734.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinbo Yu, Songlin Han","submitted_at":"2026-05-15T09:18:26Z","abstract_excerpt":"In 2013, Zhi-Wei Sun proposed a Romanov-type conjecture stating that every integer $n > 1$ can be written as $n = k + m$ with $k, m \\ge 1$ such that $2^k + m$ is a prime. In this paper, we unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least $0.0734$. We also discuss the limitations of our method. Under a uniform Hardy-Littlewood prime pairs conjecture, we show that the lower bound of density obtained by this method cannot exceed $1/(\\log 2 + 1) \\approx 0.5906$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least 0.0734.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analytic or sieve estimates used to obtain the unconditional lower bound of 0.0734 are valid and produce a strictly positive quantity (as asserted in the unconditional claim).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The set of n > 1 satisfying Sun's 2^k + m representation property has positive lower density at least 0.0734.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Natural numbers n that can be written as n = k + m with 2^k + m prime have positive asymptotic density at least 0.0734.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"03a467313349dc1d9322dece639bf17278bcedc7831ec53b4f4a82c8f22c8d57"},"source":{"id":"2605.15758","kind":"arxiv","version":1},"verdict":{"id":"d203b7e9-4f19-40bc-89e6-65066988d055","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:06:02.726941Z","strongest_claim":"We unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least 0.0734.","one_line_summary":"The set of n > 1 satisfying Sun's 2^k + m representation property has positive lower density at least 0.0734.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analytic or sieve estimates used to obtain the unconditional lower bound of 0.0734 are valid and produce a strictly positive quantity (as asserted in the unconditional claim).","pith_extraction_headline":"Natural numbers n that can be written as n = k + m with 2^k + m prime have positive asymptotic density at least 0.0734."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15758/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.659511Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:13:09.955034Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:48.764686Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:55.958289Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5386d899cd80e723f0a06eef40432a5856ecf068da21a764ab7fc7e2695ceff2"},"references":{"count":16,"sample":[{"doi":"","year":2021,"title":"Kevin Broughan,Bounded gaps between primes: The epic breakthroughs of the early twenty-first century, Cambridge University Press, Cambridge, 2021. MR 4412547","work_id":"6bb255c3-8745-47cb-9b3e-ffd68d52554d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Yong-Gao Chen and Xue-Gong Sun,On Romanoff’s constant, Journal of Number Theory106(2004), no. 2, 275–284","work_id":"7639ff02-ab68-4deb-a6cf-e6b4507f8ea2","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1971,"title":"Roger Crocker,On the sum of a prime and of two powers of two, Pacific Journal of Mathematics36 (1971), no. 1, 103–107","work_id":"89d7d704-dab8-470b-889c-d167985ad149","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Christian Elsholtz and Jan-Christoph Schlage-Puchta,On Romanov’s constant, Mathematische Zeitschrift288(2018), no. 3-4, 713–724","work_id":"ab2f48b5-06fd-4e0c-85e7-9db48259a353","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1950,"title":"Paul Erdős,On integers of the form2k +pand some related problems, Summa Brasiliensis Mathe- maticae2(1950), no. 8, 113–123","work_id":"c79a5198-1447-44a1-a9ae-6d7e93201211","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":16,"snapshot_sha256":"7c47bde3cb03df284de408cd5b978b45f8e583e8fb2c2c21d78dc24871a7e5a4","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0538f45964d83531bfd40988ce014841c66df39534eb14e855444d8823ed4e42"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}