Almost-primes in Sun's $x^2+ny^2$ conjecture
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908822366322688 |
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| author | Han, Songlin Yu, Jinbo |
| author_facet | Han, Songlin Yu, Jinbo |
| contents | In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n > 1$ admits a partition $n = x + y$ with integers $x, y >0$ such that $x + ny$ and $x^2 + ny^2$ are simultaneously prime. To approach this conjecture we use the method of weighted sieve as developed by Richert, Halberstam, and Diamond. In this article, we first formalize the conjecture into a sieve problem. We verify that the conditions required to use Richert's weighted sieve are satisfied and establish partial results with almost-prime solutions for sufficiently large $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Almost-primes in Sun's $x^2+ny^2$ conjecture Han, Songlin Yu, Jinbo Number Theory 11N32, 11N36 In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n > 1$ admits a partition $n = x + y$ with integers $x, y >0$ such that $x + ny$ and $x^2 + ny^2$ are simultaneously prime. To approach this conjecture we use the method of weighted sieve as developed by Richert, Halberstam, and Diamond. In this article, we first formalize the conjecture into a sieve problem. We verify that the conditions required to use Richert's weighted sieve are satisfied and establish partial results with almost-prime solutions for sufficiently large $n$. |
| title | Almost-primes in Sun's $x^2+ny^2$ conjecture |
| topic | Number Theory 11N32, 11N36 |
| url | https://arxiv.org/abs/2602.08286 |