Diagonal diophantine equations with small prime variables

Let (ai,aj)=1(ai,aj)=1, 1≤i0ϵ>0, we have(i)if a1,⋯,asa1,⋯,as are not all of the same sign, then the above equation has solutions in primes pjpj satisfying pj≪|n|1/k+A3⋅2k−1+ϵpj≪|n|1/k+A3⋅2k−1+ϵ,(ii)if a1,⋯,asa1,⋯,as are all positive, then the above equation is solvable in prime pjpj whenever n≫A3k⋅2k−1+1+ϵn≫A3k⋅2k−1+1+ϵ. This result is the general case of the Diophantine equations with Small Prime Variables.