Abstract
Let (ai,aj)=1, 1≤i<j≤s and s=2k+1, where a1,⋯,as,s and k≥4 are nonzero integers. In this paper, we show that if the diagonal diophantine equation a1p1k+⋯+aspsk=n is satisfying some necessary conditions, then we have the following results: For any ϵ>0, we have(i)if a1,⋯,as are not all of the same sign, then the above equation has solutions in primes pj satisfying pj≪|n|1/k+A3⋅2k−1+ϵ,(ii)if a1,⋯,as are all positive, then the above equation is solvable in prime pj whenever n≫A3k⋅2k−1+1+ϵ. This result is the general case of the Diophantine equations with Small Prime Variables.
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