On the Waring–Goldbach problem for two squares and four cubes

Min Zhang; Fei Xue; Jinjiang Li; Min Zhang; Fei Xue; Jinjiang Li

doi:10.3934/math.2022689

AIMS Mathematics

2022, Volume 7, Issue 7: 12415-12436. doi: 10.3934/math.2022689

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Research article

On the Waring–Goldbach problem for two squares and four cubes

Min Zhang ^{1
,
,},
Fei Xue ^{2
,
,},
Jinjiang Li ^{2
,
,}

1.
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
2.
Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

Received: 04 December 2021 Revised: 18 April 2022 Accepted: 18 April 2022 Published: 26 April 2022
MSC : 11N36, 11P05, 11P32, 11P55

Let $ \mathcal{P}_r $ denote an almost–prime with at most $ r $ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $ N $, the following equation

$ \begin{equation*} N = p_1^2+p_2^2+x^3+p_3^3+p_4^3+p_5^3 \end{equation*} $

is solvable with $ x $ being an almost–prime $ \mathcal{P}_7 $ and the other variables primes. This result constitutes a deepening upon that of previous results.
- Waring–Goldbach problem,
- Hardy–Littlewood method,
- sieve method,
- almost–prime
Citation: Min Zhang, Fei Xue, Jinjiang Li. On the Waring–Goldbach problem for two squares and four cubes[J]. AIMS Mathematics, 2022, 7(7): 12415-12436. doi: 10.3934/math.2022689

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Abstract

Let $ \mathcal{P}_r $ denote an almost–prime with at most $ r $ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $ N $, the following equation

$ \begin{equation*} N = p_1^2+p_2^2+x^3+p_3^3+p_4^3+p_5^3 \end{equation*} $

is solvable with $ x $ being an almost–prime $ \mathcal{P}_7 $ and the other variables primes. This result constitutes a deepening upon that of previous results.

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