Geodesic
On the Waring-Goldbach problem for one square and five cubes in short intervals
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 563-589
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Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$ n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3, \Bigl | p_1^2-\dfrac {N}{6}\Bigr | \leq U, \quad \Bigl | p_i^3-\dfrac {N}{6}\Bigr |\leq U, \quad i=2,3,4,5,6, $$ where $U=N^{1-\delta +\varepsilon }$ with $\delta \leq 8/225$.
Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$ n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3, \Bigl | p_1^2-\dfrac {N}{6}\Bigr | \leq U, \quad \Bigl | p_i^3-\dfrac {N}{6}\Bigr |\leq U, \quad i=2,3,4,5,6, $$ where $U=N^{1-\delta +\varepsilon }$ with $\delta \leq 8/225$.
DOI : 10.21136/CMJ.2020.0013-20
Classification : 11P05, 11P32, 11P55
Keywords: Waring-Goldbach problem; Hardy-Littlewood method; exponential sum; short interval 
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Xue, Fei; Zhang, Min; Li, Jinjiang. On the Waring-Goldbach problem for one square and five cubes in short intervals. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 563-589. doi: 10.21136/CMJ.2020.0013-20

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