The exceptional set for Diophantine inequality with mixed powers of primes
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915953244110848 |
|---|---|
| author | Fu, Yu Fu, Linzhu Hu, Liqun |
| author_facet | Fu, Yu Fu, Linzhu Hu, Liqun |
| contents | Assume that $λ_1, λ_2, λ_3,λ_4,λ_5,λ_6,λ_7$ are non-zero real numbers , $λ_1/λ_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $δ>0$. For any given positive integer $k\geq 5$ and any $\varepsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V} $ with $\upsilon \leq X$ for which the inequality $$ \left | λ_1p_1^2 + λ_2p_2^3 + λ_3p_3^3 + λ_4p_4^3 + λ_5p_5^3 + λ_6p_6^4 + λ_7p_7^k - \upsilon \right | <{\upsilon}^{-δ} $$ has no solution in primes $p_1, p_2, p_3, p_4, p_5, p_6, p_7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22147 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The exceptional set for Diophantine inequality with mixed powers of primes Fu, Yu Fu, Linzhu Hu, Liqun Number Theory Assume that $λ_1, λ_2, λ_3,λ_4,λ_5,λ_6,λ_7$ are non-zero real numbers , $λ_1/λ_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $δ>0$. For any given positive integer $k\geq 5$ and any $\varepsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V} $ with $\upsilon \leq X$ for which the inequality $$ \left | λ_1p_1^2 + λ_2p_2^3 + λ_3p_3^3 + λ_4p_4^3 + λ_5p_5^3 + λ_6p_6^4 + λ_7p_7^k - \upsilon \right | <{\upsilon}^{-δ} $$ has no solution in primes $p_1, p_2, p_3, p_4, p_5, p_6, p_7$. |
| title | The exceptional set for Diophantine inequality with mixed powers of primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.22147 |