Diophantine Inequality With Prime Variables And Mixed Powers

In the paper,we mainly discuss the Diophantine approach problem with one prime,one square,one cubed and one k-th power of a prime.Using Davenport-Heilbronn’s improved Hardy-Littlewood method,the Diophantine approach problem is transformed into the integral problem.The integral interval is divided into three disjoint intervals.There are the major arc,the minor arc and the trivial arc.Then the prime variable triangle is used to obtain the interval.Finally,we prove the following theorem.Theorem 1.Assume that 1<k<16/7,λ1,λ2,λ3m,λ4 are non-zero real numbers,not all of the same sign,that λ1/λ2 is irrational and letη be a real number.The inequality{λ1p1+λ2p22+λ3p33+λ4p4k+η|≤(max pj)-(16-7k)/16k+εhas infinitely many solutions in primes variables for any ε>0.