| Title:
|
A Diophantine inequality with four squares and one $k$th power of primes (English) |
| Author:
|
Mu, Quanwu |
| Author:
|
Zhu, Minhui |
| Author:
|
Li, Ping |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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69 |
| Issue:
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2 |
| Year:
|
2019 |
| Pages:
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353-363 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $k\geq 5$ be an odd integer and $\eta $ be any given real number. We prove that if $\lambda _1$, $\lambda _2$, $\lambda _3$, $\lambda _4$, $\mu $ are nonzero real numbers, not all of the same sign, and $\lambda _1/\lambda _2$ is irrational, then for any real number $\sigma $ with $0<\sigma <1/(8\vartheta (k))$, the inequality $$ |\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+ \eta |<\Bigl (\max _{1\leq j\leq 5} p_j\Bigr )^{-\sigma } $$ has infinitely many solutions in prime variables $p_1, p_2, \cdots , p_5$, where $\vartheta (k)=3\times 2^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta (k)=[(k^2+2k+5)/8]$ for odd integer $k$ with $k\geq 11$. This improves a recent result in W. Ge, T. Wang (2018). (English) |
| Keyword:
|
Diophantine inequalities |
| Keyword:
|
Davenport-Heilbronn method |
| Keyword:
|
prime |
| MSC:
|
11D75 |
| MSC:
|
11P55 |
| idZBL:
|
Zbl 07088789 |
| idMR:
|
MR3959949 |
| DOI:
|
10.21136/CMJ.2018.0316-17 |
| . |
| Date available:
|
2019-05-24T08:55:46Z |
| Last updated:
|
2021-07-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147729 |
| . |
| Reference:
|
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