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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 $1/(\\log 2 + 1) \\approx 0.5906$.","authors_text":"Jinbo Yu, Songlin Han","cross_cats":[],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-15T09:18:26Z","title":"Positive density for Sun's $2^k+m$ conjecture"},"references":{"count":16,"internal_anchors":1,"resolved_work":16,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Kevin Broughan,Bounded gaps between primes: The epic breakthroughs of the early twenty-first century, Cambridge University Press, Cambridge, 2021. 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