A Diophantine inequality with four squares and one $k$th power of primes

Quanwu Mu; Minhui Zhu; Ping Li

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 2, page 353-363
  • ISSN: 0011-4642

Abstract

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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).

How to cite

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Mu, Quanwu, Zhu, Minhui, and Li, Ping. "A Diophantine inequality with four squares and one $k$th power of primes." Czechoslovak Mathematical Journal 69.2 (2019): 353-363. <http://eudml.org/doc/294523>.

@article{Mu2019,
abstract = {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).},
author = {Mu, Quanwu, Zhu, Minhui, Li, Ping},
journal = {Czechoslovak Mathematical Journal},
language = {eng},
number = {2},
pages = {353-363},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A Diophantine inequality with four squares and one $k$th power of primes},
url = {http://eudml.org/doc/294523},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Mu, Quanwu
AU - Zhu, Minhui
AU - Li, Ping
TI - A Diophantine inequality with four squares and one $k$th power of primes
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 353
EP - 363
AB - 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).
LA - eng
UR - http://eudml.org/doc/294523
ER -

References

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