素变量混合幂丢番图方程

Diophantine Equation with Prime Variables Mixed Power

设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+λ2p22+λ3p33+λ4p34+η<(maxpj)^-σ有无穷多组素数解p1,p2,p3,p4,使用Davenport-Heilbronn方法计算,得到maxpj的指数估计为σ=7/16+ε,ε>0.

Let η be any real number, supposing that λ 1,λ 2,λ 3,λ 4 are non-zero 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+λ 2p 2 2+λ 3p 3 3+λ 4p k 4+η<(max p j)^-σ has many infinitely solutions in prime variables p 1,p 2,p 3,p 4 .Using Davenport-Heilbronn method, the index max p j is σ= 7/16 +ε,ε>0.