Positive density for Sun's $2^k+m$ conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917499686092800 |
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| author | Han, Songlin Yu, Jinbo |
| author_facet | Han, Songlin Yu, Jinbo |
| contents | 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 $1/(\log 2 + 1) \approx 0.5906$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_15758 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive density for Sun's $2^k+m$ conjecture Han, Songlin Yu, Jinbo Number Theory 11N32 (Primary), 11N36, 11P32 (Secondary) 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 $1/(\log 2 + 1) \approx 0.5906$. |
| title | Positive density for Sun's $2^k+m$ conjecture |
| topic | Number Theory 11N32 (Primary), 11N36, 11P32 (Secondary) |
| url | https://arxiv.org/abs/2605.15758 |