素变量混合幂丢番图逼近

Diophantine Approximation with Prime Variables and Mixed Power

设k是大于或等于的正整数,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,并且λ_1/λ_2是无理数,则不等式|λ_1p_1~2+λ_2p_2~2+λ_3p_3~2+λ_4p_4~k+η|<(maxp_j)^(-σ)有无穷多组素数解p_1,p_2,p_3,p_4,这里σ=1/8(k+8/k)+ε,ε>0.

Let k be an integer with k≥3 andηbe any real number.Suppose that λ_1,λ_2,λ_3,λ_4 are nonzero real numbers,not all of them have the same sign and λ_1/λ_2 is irrational.It is proved that the inequality |λ_1p_1~2+λ_2p_2~2+λ_3p_3~2+λ_4p_4~k+η|<(maxp_j)^(-σ) has infinitely many solutions in prime variables p_1,p_2,p_3,p_4.Here σ=1/8(k+8/k)+ε,ε>0.