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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.07146 |
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| _version_ | 1866910147926818816 |
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| author | Long, Linji Li, Jinjiang Zhang, Min Sun, Rui |
| author_facet | Long, Linji Li, Jinjiang Zhang, Min Sun, Rui |
| contents | Suppose that $c,d,α,β$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<α<β<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $α\leqslant N_2/N_1^{d/c}\leqslantβ$, the following Diophantine inequalities system \begin{align*} \begin{cases} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N_1|<\varepsilon_1 (N_1) \\ |p_1^d+p_2^d+p_3^d+p_4^d+p_5^d+p_6^d-N_2|<\varepsilon_2 (N_2) \end{cases} \end{align*} is solvable in prime variables $p_1, p_2, p_3, p_4, p_5, p_6$, where \begin{align*} \begin{cases} \varepsilon_1 (N_1)=N_1^{-(1/c)(79/71-c)} (\log N_1)^{201}, \\ \varepsilon_2 (N_2)=N_2^{-(1/d)(79/71-d)} (\log N_2)^{201} . \end{cases} \end{align*} This result constitutes an improvement upon the previous result of Han-Liu-Zhang [5]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a system of two Diophantine inequalities with six prime variables Long, Linji Li, Jinjiang Zhang, Min Sun, Rui Number Theory Suppose that $c,d,α,β$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<α<β<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $α\leqslant N_2/N_1^{d/c}\leqslantβ$, the following Diophantine inequalities system \begin{align*} \begin{cases} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N_1|<\varepsilon_1 (N_1) \\ |p_1^d+p_2^d+p_3^d+p_4^d+p_5^d+p_6^d-N_2|<\varepsilon_2 (N_2) \end{cases} \end{align*} is solvable in prime variables $p_1, p_2, p_3, p_4, p_5, p_6$, where \begin{align*} \begin{cases} \varepsilon_1 (N_1)=N_1^{-(1/c)(79/71-c)} (\log N_1)^{201}, \\ \varepsilon_2 (N_2)=N_2^{-(1/d)(79/71-d)} (\log N_2)^{201} . \end{cases} \end{align*} This result constitutes an improvement upon the previous result of Han-Liu-Zhang [5]. |
| title | On a system of two Diophantine inequalities with six prime variables |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.07146 |