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 ≥ 5 be an odd integer and η be any given real number. We prove that if λ 1 , λ 2 , λ 3 , λ 4 , μ are nonzero real numbers, not all of the same sign, and λ 1 / λ 2 is irrational, then for any real number σ with 0 < σ < 1 / ( 8 ϑ ( k ) ) , the inequality | λ 1 p 1 2 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 2 + μ p 5 k + η | < max 1 ≤ j ≤ 5 p j - σ has infinitely many solutions in prime variables p 1 , p 2 , ⋯ , p 5 , where ϑ ( k ) = 3 × 2 ( k - 5 ) / 2 for k = 5 , 7 , 9 and ϑ ( k ) = [ ( k 2 + 2 k + 5 ) / 8 ] for odd integer k with k ≥ 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 $k\ge 5$ 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 \[ |\lambda \_1p\_1^2+\lambda \_2p\_2^2+\lambda \_3p\_3^2+\lambda \_4p\_4^2+\mu p\_5^k+ \eta |<\Bigl (\max \_\{1\le j\le 5\} p\_j\Bigr )^\{-\sigma \} \] 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 $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).},
author = {Mu, Quanwu, Zhu, Minhui, Li, Ping},
journal = {Czechoslovak Mathematical Journal},
keywords = {Diophantine inequalities; Davenport-Heilbronn method; prime},
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 $k\ge 5$ 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 \[ |\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+ \eta |<\Bigl (\max _{1\le j\le 5} p_j\Bigr )^{-\sigma } \] 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 $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).
LA - eng
KW - Diophantine inequalities; Davenport-Heilbronn method; prime
UR - http://eudml.org/doc/294523
ER -

References

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