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 ? is an any real number.Supposed that?₁,?₂,?₃ are nonzero real numbers,not all of the same sign,with?₁/?₂ is irrational.It is proved that the inequality |?₁p₁+?₂p₂+?₃p₃^k+?|??maxp_j?^? has infinitely many solutions in prime variables p₁,p₂,p₃,where0???1/66 for k=4 and 0???2^k-1/(2^k+1k?k+1? for k?5.