摘要
运用Davenport-Heilbronn方法证明了:如果μ1, . . . , μr是不全为负的非零实数,至少一个μj (1≤ j ≤ r)是无理数, k,m,r是正整数, k ≥ 4, r ≥ 2k-1+1,则存在无穷多素数p1, . . . , pr, p,使得[μ1p1k+···+μrprk]=mp.. 特别地,[μ1p1k+···+μrprk]可表示无穷多素数.
Abstract
Using Davenport-Heilbronn method, this paper shows that if μ1, . . . , μr are non-zero real numbers, not both negative, at least one of μj (1≤ j ≤ r) is irrational, and k,m, r are positive integers satisfying k ≥ 4, r ≥ 2k-1+1, then there exist infinitely many primes p1, . . . , pr, p such that [μ1p1k+···+μrprk]=mp. In particular, [μ1p1k+···+μrprk] represents infinitely many primes.
关键词
素数变量 /
丢番图逼近 /
Davenport-Heilbronn方法
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Key words
prime variables /
diophantine approximation /
Davenport-Heilbronn method
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李伟平, 王天泽.
素数k次方和的非线性型的整数部分. 数学学报, 2013, 56(4): 605-612 https://doi.org/10.12386/A2013sxxb0061
Wei Ping LI, Tian Ze WANG.
The Integral Part of a Nonlinear Form with k Powers of Primes. Acta Mathematica Sinica, Chinese Series, 2013, 56(4): 605-612 https://doi.org/10.12386/A2013sxxb0061
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参考文献
[1] Danicic I., On the integral part of a linear form with prime variables, Canadian J. Math., 1966, 18: 621-628.
[2] Davenport H., Heilbronn H., On indefinite quadratic forms in five variables, Journal of London Math. Soc., 1946, 21: 185-193.
[3] Harman G., Trigonometric sums over primes I, Mathematika, 1981, 28: 249-254.
[4] Li W. P., Wang T. Z., The integral part of a nonlinear form with three squares of primes, Chinese Annals of Mathematics, 2011, 32(6): 753-762.
[5] Vaughan R. C., Diophantine approximation by prime numbers (I), Proc. London Math. Soc., 1974, 28: 373-384.
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脚注
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基金
国家自然科学基金资助项目(11071070);河南省教育厅自然科学研究计划(2011B110002)
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