Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...,x<sub>7</sub> such that the inequalities |λ<sub>1</sub>x<sub>1</sub><sup>3</sup>+λ<sub>2</sub>x<sub>2</sub><sup>3</sup>+...+λ<sub>7</sub>x<sub>7</sub><sup>3</sup>|【1 and 0【sum from i=1 to7(λ<sub>i</sub>|x<sub>i</sub>]<sup>3</sup> （λ<sub>1</sub>λ<sub>2</sub>…λ<sub>7</sub>）<sup>89814</sup>) hold simultaneously.展开更多
For arbitrary c00, if A is a subset of the primes less than x with cardinality δx（logx）-1, δ≥（logx）-c0, then there exists a positive constant c such that the cardinality of A＋A is larger than cδx（loglogx）-1.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19671051)
文摘Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...,x<sub>7</sub> such that the inequalities |λ<sub>1</sub>x<sub>1</sub><sup>3</sup>+λ<sub>2</sub>x<sub>2</sub><sup>3</sup>+...+λ<sub>7</sub>x<sub>7</sub><sup>3</sup>|【1 and 0【sum from i=1 to7(λ<sub>i</sub>|x<sub>i</sub>]<sup>3</sup> （λ<sub>1</sub>λ<sub>2</sub>…λ<sub>7</sub>）<sup>89814</sup>) hold simultaneously.
基金Supported by National Natural Science Foundation of China (Grant No.11271249)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20120073110059)
文摘For arbitrary c00, if A is a subset of the primes less than x with cardinality δx（logx）-1, δ≥（logx）-c0, then there exists a positive constant c such that the cardinality of A＋A is larger than cδx（loglogx）-1.
