Diophantine Inequalities with Mixed Powers

MU Quan-wu, LÄU Xiao-dong

doi:10.13371/j.cnki.chin.q.j.m.2015.04.007

Chinese Quarterly Journal of Mathematics >

2015 , Vol. 30 >Issue 4: 545 - 554

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2015.04.007

Diophantine Inequalities with Mixed Powers

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1. College of Science, Xi'an Polytechnic University2. Department of Mathematics, Tongji University

MU Quan-wu(1977{), male, native of Yangxian, Shaanxi, a lecturer of Xi'an Polytechnic University, Ph.D., engages in analytic number theory; LÄU Xiao-dong(1989{), male, native of Linyi, Shandong, Ph.D. student, engages in analytic number theory.

Received date: 2014-01-08

Online published: 2020-11-19

Supported by

Supported by the National Natural Science Foundation of China(11201107,11271283,11501435)； Supported by the Natural Science Foundation of Anhui Province(1208085QA01)；

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Abstract

It is proved that if λ₁, λ₂, ···, λ₇are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 < σ <1/16, the inequality |λ₁x₁²+ λ₂x₂²+∑7 i=3λ_ix_i⁴+ η| <（ max1≤i≤7|x_i|）^-σhas infinitely many solutions in positive integers λ₁, λ₂, ···, λ₇ Similar result is proved for |λ₁x₁²+ λ₂x₂²+ λ₃x₃²+ λ₄x₄⁴+ λ₅x₅⁴+ λ₆x₆⁴+ η| <（ max1≤i≤6|x_i|）^-σ.These results constitute an improvement upon those of Shi and Li.

Key words： diophantine inequality; mixed power; the Davenport-Heilbronn method

Cite this article

MU Quan-wu, LÄU Xiao-dong . Diophantine Inequalities with Mixed Powers[J]. Chinese Quarterly Journal of Mathematics, 2015 , 30(4) : 545 -554 . DOI: 10.13371/j.cnki.chin.q.j.m.2015.04.007

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