一个素变量混合幂丢番图不等式

A Diophantine inequality with prime variables and mixed power

目的证明素数p_j对不等式︱λ_1p_1+λ_2p_2~2+λ_3p_3~2+λ_4p_4~k-v︱<(maxp_j)^(-1/8σ(k)+ε)有无穷多个解,其中k是大于或等于3的正整数,ε>0,v是任意给定的实数,σ(k)=min(2^(s(k)-1),1/2(s(k)+1)),s(k)=[k+1/2],假设λ_1,λ_2,λ_3,λ_4是非零的实数,并且λ_1/λ_2是无理数。方法使用Davenport-Heilbronn方法来改进这一结果。结果与结论 maxp_j的指数估计为-1/8σ(k)+ε。

Purpose- To prove that there are infinitely many solutions to the inequality︱λ1p1＋λ2p2-2＋λ3p3-2＋λ4p4-k-v︱〈（maxpj）^（-1/8σ（k）＋ε） relative to primes pimes pj,where k is a positive integer which is greater than or equal to 3,ε is greater than 0 and v is any given real number, by letting σ（k）=min（2^（s（k）-1）,1/2（s（k）＋1））,where s（k）=[k＋1/2], and supposing that λ1,λ2,λ3, and λ4 are nont-zero real numbersand λ1/λ2 is irrational number. Methods-Davenportt-Heilbronn method is used to improve the results. Results and Conclusion-The index of max pj is estimated to be-1/8σ（k）＋ε.