混合幂为1,1,3和k的丢番图不等式

Diophantine Inequality with Mixed Powers 1,1,3 and k

设k 是大于或等于3的正整数,η是任意给定的实数,假设λ1,λ2,λ3,λ4 是非零实数,不全同号,并且λ1/λ2 是无理数,则不等式|λ1p1 +λ2p2 +λ3p33+λ4pk4+η|<(maxpj)-σ有无穷多组素数解p1,p2,p3,p4,这里σ= 116k +4 k ()+ε,ε>0.

Let k be an positive integer with k≥3 andηbe any real number.Supposing thatλ1,λ2,λ3,λ4 are non-zero real numbers,not all of them have the same sign andλ1/λ2 is irrational.It is proved that the inequality|λ1p 1+λ2p 2+λ3p 3 3+λ4p k 4+η|<(max p j)-σhas infinitely many solutions in prime variables p 1,p 2,p 3,p 4,whereσ=1 16 k+4 k+ε,ε>0.