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Keywords:
Waring-Goldbach problem; exponential sum over prime in short interval; circle method
Waring-Goldbach problem; exponential sum over prime in short interval; circle method
Summary:
We prove that almost all positive even integers $n$ can be represented as $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ with $|p_{k}^{k}-\tfrac 14 N|\leq N^{1-1/54+\varepsilon }$ for $2\leq k\leq 5$. As a consequence, we show that each sufficiently large odd integer $N$ can be written as $p_{1}+p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ with $|p_{k}^{k}- \tfrac 15 N|\leq N^{1-1/54+\varepsilon }$ for $1\leq k\leq 5$.
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