Geodesic
On an additive problem of unlike powers in short intervals
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1167-1174
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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 $1\leq k\leq 5$.
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 $1\leq k\leq 5$.
DOI : 10.21136/CMJ.2022.0417-21
Classification : 11P05, 11P32, 11P55
Keywords: Waring-Goldbach problem; exponential sum over prime in short interval; circle method 
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Zhang, Qingqing. On an additive problem of unlike powers in short intervals. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1167-1174. doi: 10.21136/CMJ.2022.0417-21

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