A Diophantine Problem with Unlike Powers of Primes

Abstract

Let k be an integer with 4≤k≤6 and η be any real number. Suppose that λ1,λ2,…,λ5 are nonzero real numbers, not all of them have the same sign, and λ1/λ2 is irrational. It is proved that the inequality λ1p1+λ2p22+λ3p33+λ4p44+λ5p5k+η<max1≤j≤5pj−σk has infinitely many solutions in prime variables p1,p2,p3,p4, and p5, where 0<σ4<1/36,0<σ5<4/189, and 0<σ6<1/54. This gives an improvement of the recent results.