Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime
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arXiv
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866914848283033600 |
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| author | Gambini, Alessandro |
| author_facet | Gambini, Alessandro |
| contents | Let $1<k<14/5$, $λ_1,λ_2,λ_3$ and $λ_4$ be non-zero real numbers, not all of the same sign such that $λ_1/λ_2$ is irrational and let $ω$ be a real number. We prove that the inequality $|λ_1p_1+λ_2p_2^2+λ_3p_3^2+λ_4p_4^k-ω|\le (\max_j p_j)^{-\frac{14-5k}{28k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1703_02381 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime Gambini, Alessandro Number Theory 11D75 (Primary), 11J25, 11P32, 11P55 (Secondary) Let $1<k<14/5$, $λ_1,λ_2,λ_3$ and $λ_4$ be non-zero real numbers, not all of the same sign such that $λ_1/λ_2$ is irrational and let $ω$ be a real number. We prove that the inequality $|λ_1p_1+λ_2p_2^2+λ_3p_3^2+λ_4p_4^k-ω|\le (\max_j p_j)^{-\frac{14-5k}{28k}+\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\varepsilon>0$. |
| title | Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime |
| topic | Number Theory 11D75 (Primary), 11J25, 11P32, 11P55 (Secondary) |
| url | https://arxiv.org/abs/1703.02381 |