混合幂丢番图不等式

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

数学季刊 ›› 2015, Vol. 30 ›› Issue (4): 545-554.doi: 10.13371/j.cnki.chin.q.j.m.2015.04.007

混合幂丢番图不等式

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

收稿日期:2014-01-08 出版日期:2015-12-30 发布日期:2020-11-19
作者简介: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.
基金资助:
Supported by the National Natural Science Foundation of China(11201107,11271283,11501435)； Supported by the Natural Science Foundation of Anhui Province(1208085QA01)；

Diophantine Inequalities with Mixed Powers

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

Received:2014-01-08 Online:2015-12-30 Published:2020-11-19
About author: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.
Supported by:
Supported by the National Natural Science Foundation of China(11201107,11271283,11501435)； Supported by the Natural Science Foundation of Anhui Province(1208085QA01)；

摘要/Abstract

摘要： 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.

关键词: diophantine inequality, mixed power, the Davenport-Heilbronn method

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

中图分类号:

O156.7

牟全武, 吕晓东. 混合幂丢番图不等式[J]. 数学季刊, 2015, 30(4): 545-554.

MU Quan-wu, LÄU Xiao-dong. Diophantine Inequalities with Mixed Powers[J]. Chinese Quarterly Journal of Mathematics, 2015, 30(4): 545-554.

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[2]	龚克, 李伟平.. 一个混合幕为2和4的丢番图不等式[J]. 数学季刊, 2008, 23(1): 1-7.

混合幂丢番图不等式

Diophantine Inequalities with Mixed Powers

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