幂次为2,3和4的整变量非线性型的整数部分

李伟平, 苏白云, 王天泽

数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 273-280.

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数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 273-280. DOI: 10.12386/A2014sxxb0027
论文

幂次为2,3和4的整变量非线性型的整数部分

    李伟平1, 苏白云1, 王天泽2
作者信息 +

The Integer Parts of a Nonlinear Form with Integer Variables and Powers 2, 3 and 4

    Wei Ping LI1, Bai Yun SU1, Tian Ze WANG2
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文章历史 +

摘要

假设λ1λ2λ3λ4是正实数,λi/λj(1 ≤ i < j ≤ 4)至少有一个是无理数.那么,对于正整数x1x2x3x4λ1x12+λ2x23+λ3x34+λ4x44的整数部分可表示无穷多素数. 这个证明极大改进了以前的结果.

Abstract

We prove that if λ1, λ2, λ3, λ4 are positive real numbers, and at least one of λi/λj (1 ≤ i < j ≤ 4) is irrational, then, the integer parts of λ1x12 +λ2x23 +λ3x34 +λ4x44 are prime infinitely often for natural numbers x1, x2, x3, x4. This greatly improves of the former result.

关键词

整数变量 / 丢番图逼近 / Davenport-Heilbronn方法

Key words

integer variables / diophantine approximation / Davenport-Heilbronn method

引用本文

导出引用
李伟平, 苏白云, 王天泽. 幂次为2,3和4的整变量非线性型的整数部分. 数学学报, 2014, 57(2): 273-280 https://doi.org/10.12386/A2014sxxb0027
Wei Ping LI, Bai Yun SU, Tian Ze WANG. The Integer Parts of a Nonlinear Form with Integer Variables and Powers 2, 3 and 4. Acta Mathematica Sinica, Chinese Series, 2014, 57(2): 273-280 https://doi.org/10.12386/A2014sxxb0027

参考文献

[1] Brüdern J., Kawada K., Wooley T. D., Additive Representation in thin Sequences, Ⅷ: Diophantine Inequalities in Review, Series on Number Theory and Its Applications, 2010, 6: 20-79.
[2] Davenport H., Roth K. F., The solubility of certain diophantine inequalities, Mathematika, 1955, 2: 81-96.
[3] LiW. P., Zhao F., Wang T. Z., The integer parts of nonlinear form with integer variables and mixed powers 2 and 3, Acta Mathematica Sinica, Chinese Series, 2012, 55(4): 727-736.
[4] Vaughan R. C., Diophantine approximation by prime numbers, Ⅰ, Proc. London Math. Soc., 1974, 28: 373-384.
[5] Vaughan R. C., Diophantine approximation by prime numbers, Ⅱ. Proc. London Math. Soc., 1974, 28: 385-401.
[6] Vaughan R. C., The Hardy-Littlewood Method, Second Edition, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997, Vol.125.

基金

国家自然科学基金项目(11071070);河南省教育厅自然科学研究计划(2011B110002)及河南省创新型科技人才队伍建设工程
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