素变量混合幂丢番图方程

目的研究素变量p_j对不等式|λ_1p_1+λ_2p_2~2+λ_3p_3~3+λ_4p_4~k+η|≤(maxp_j)~(-σ)有无穷多组素数解时的情况下σ的取值,其中1k24/13,η是任意给定的实数,λ_1,λ_2,λ_3,λ_4是非零实数不全同号,且λ_1/λ_2是无理数。方法使用Davenport-Heilbronn方法来计算。结果与结论得到maxp_j的指数估计为σ=1/48(24-13k/k)+ε,ε0。

Diophantine equation with mixed powers and prime variables

Purposes—To investigate the value of σ when prime variable pjmeets the fact that the inequality|λ1p1+λ2p22+λ3p33+λ4pk4+η|≤(max pj)-σhas infinitely many solutions if k is let to satisfy 1 < k < 24/13 and η to be any real number and it is supposed that λ1,λ2,λ3and λ4are non-zero real numbers with incomplete same sign and λ1/λ2is irrational numbers.Methods—Davenport-Heilbronn method is used to make the calculation.Results and Conclusion—The index max pjis estimated to beσ= 1/48 (24 -13 k/ k)+ εfor any εwhich is greater than zero.