Study On Diophantine Approximation With Two Primes And K-power Of A Prime

In this paper,we mainly study the problem of Diophantine approximation with two primes and k-power of a prime.On the basis of Mu Quanwu’s research,the result of the Diophantine problem is improved by using the method of enlarged major arc.The conclusions are as follows: Let k be an integer with k≥4 and η is an any real number.Supposed thatλ₁,λ₂,λ₃ are nonzero real numbers,not all of the same sign,withλ₁/λ₂ is irrational.It is proved that the inequality |λ₁p₁+λ₂p₂+λ₃p₃^k+η|＜（maxp_j）^σ has infinitely many solutions in prime variables p₁,p₂,p₃,where0＜σ＜1/66 for k=4 and 0＜σ＜2^k-1/(2^k+1k（k+1） for k≥5.