Let N be a sufficiently large integer. We prove that almost all sufficiently large even integers n ∈ [N − 6U, N + 6U] can be represented as

$$\left\{ {\matrix{ {n = p_1^2 + p_2^3 + p_3^3+ p_4^3 + p_5^3 + p_6^3,} \hfill \;\;\; {} \hfill \cr {\left| {p_1^2 - {N \over 6}} \right| \leqslant U,\,\,\,\,\,\left| {p_i^3 - {N \over 6}} \right| \leqslant U,\,\,} \hfill \;\;\; {i = 2,3,4,5,6,} \hfill \cr } } \right.$$

where U = N^1−δ+ε with δ ⩽ 8/225.

中文翻译：

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关于一平方和五个立方的短时间间隔的Waring-Goldbach问题

令N为足够大的整数。我们证明了几乎所有的足够大的偶数Ñ ∈[ Ñ - 6 ü，N + 6 ù ]可被表示为

$$ \ left \ {{\ matrix {{n = p_1 ^ 2 + p_2 ^ 3 + p_3 ^ 3 + p_4 ^ 3 + p_5 ^ 3 + p_6 ^ 3，} \ hfill \; \; \; {} \ hfill \ cr {\ left | {p_1 ^ 2-{N \ over 6}} \ right | \ leqslant U，\，\，\，\，\，\ left | {p_i ^ 3-{N \ over 6}} \ right | \ leqslant U，\，\，} \ hfill \; \; \; {i = 2,3,4,5,6，} \ hfill \ cr}} \ right。$$

其中û = Ñ ^{1- δ+ε}与δ ⩽225分之8。