Diagonal diophantine equations with small prime variables

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Let (a_i,a_j)=1

(a_{i}, a_{j}) = 1

, 1≤i<j≤s

1 \leq i < j \leq s

and s=2^k+1

s = 2^{k} + 1

, where a₁,⋯,a_s,s

a_{1}, \dots, a_{s}, s

and k≥4

k \geq 4

are nonzero integers. In this paper, we show that if the diagonal diophantine equation

a_{1} p_{1}^{k} + \dots + a_{s} p_{s}^{k} = n

is satisfying some necessary conditions, then we have the following results: For any ϵ>0

ϵ > 0

, we have

(i): if a₁,⋯,a_s $a_{1}, \dots, a_{s}$ are not all of the same sign, then the above equation has solutions in primes p_j $p_{j}$ satisfying p_j≪|n|^1/k+A^{3⋅2^k−1+ϵ} $p_{j} ≪ | n |^{1 / k} + A^{3 \cdot 2^{k - 1} + ϵ}$ ,
(ii): if a₁,⋯,a_s $a_{1}, \dots, a_{s}$ are all positive, then the above equation is solvable in prime p_j $p_{j}$ whenever n≫A^{3k⋅2^k−1+1+ϵ} $n ≫ A^{3 k \cdot 2^{k - 1} + 1 + ϵ}$ .

This result is the general case of the Diophantine equations with Small Prime Variables.