Diophantine Inequality With Prime Variables And Mixed Powers

In this paper,we mainly study the Diophantine inequality problem with one prime,one square of prime,two cubed of primes and one k-th power of a prime,where k is a real number or an integer,by means of Davenport-Heilbronn’s improved Hardy-Littlewood method,we obtain the following results.Theorem 1 Assume that k₁ is a real number and 1<k₁<8/3,λ₁,λ₂,λ₃,λ₄,λ₅ are non-zero real numbers,not all of the same sign,that λ₁/λ₂ is irrational and let η be a real number.The inequality |λ₁p₁+λ₂p₂²+λ₃p₃³+λ₄p₄³+λ₅p₅^k₁+η|<（maxp_j）^{-1/16（8-3k₁/k₁）+ε} has infinitely many solutions in primes variables p₁,p₂,p₃,p₄,p₅ for any ε>0.Theorem 2 Assume that k₂ is an integer and k₂≥3,λ₁,λ₂,λ₃,λ₄,λ₅ are non-zero real numbers,not all of the same sign,that λ₁/λ₂ is irrational and let η be a real number σ（k₂）=min(2^{s（k₂）-1},1/2s（k₂）（s（k₂）+1）),s（k₂）=[k₂+1/2],,The inequality|λ₁p₁+λ₂p₂²+λ₃p-3³+λ₄p₄³+λ₅p₅^k₂+η|<（max p_j）^{-1/16σ（k₂）+ε} has infinitely many solutions in primes variables p₁,p₂,p₃,p₄,p₅ for any ε>0.