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On the Waring-Goldbach problem for one square and five cubes in short intervals. (English) Zbl 07361086

Summary: 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 \[\begin{cases}n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\\ \Bigl | p_1^2-\dfrac{N}{6}\Bigr | \leq U,\quad\Bigl | p_i^3-\dfrac{N}{6}\Bigr |\leq U,\quad i=2,3,4,5,6,\end{cases}\] where \(U=N^{1-\delta +\varepsilon}\) with \(\delta\leq 8/225\).

MSC:

11P05 Waring’s problem and variants
11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
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