On Waring–Goldbach problem of mixed powers

Xiaodong Lü; Quanwu Mu

doi:10.5802/jtnb.951

Xiaodong Lü ¹; Quanwu Mu ²

¹ Department of Mathematics Tongji University Shanghai, 200092, P. R. China
² School of Science Xi’an Polytechnic University Xi’an, Shaanxi Province, 710048, P. R. China

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 523-538.

Abstract
Résumé

Let $P_{r}$ 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 odd integer $N$ , the equation

N = x^{2} + p_{1}^{3} + p_{2}^{3} + p_{3}^{3} + p_{4}^{3} + p_{5}^{6} + p_{6}^{7}

is solvable with $x$ being an almost-prime $P_{42}$ and the other terms powers of primes.

Un nombre presque premier est un $P_{r}$ s’il a au plus $r$ facteurs premiers, comptés avec multiplicité. Dans cet article nous montrons que pour tout entier impair $N$ suffisamment grand, l’équation

N = x^{2} + p_{1}^{3} + p_{2}^{3} + p_{3}^{3} + p_{4}^{3} + p_{5}^{6} + p_{6}^{7}

admet une solution avec $x$ un nombre presque premier $P_{42}$ et les autres termes étant des puissances de nombres premiers.

Received: 2014-05-18
Revised: 2015-02-27
Accepted: 2015-04-14
Published online: 2017-01-02

MR: 3509722 | Zbl: 1415.11129

DOI: 10.5802/jtnb.951

Classification: 11P32, 11N36
Keywords: Waring–Goldbach problem, circle method, sieve method, almost-prime

Author's affiliations:

Xiaodong Lü ¹; Quanwu Mu ²

¹ Department of Mathematics Tongji University Shanghai, 200092, P. R. China
² School of Science Xi’an Polytechnic University Xi’an, Shaanxi Province, 710048, P. R. China

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     title = {On {Waring{\textendash}Goldbach} problem of mixed powers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {523--538},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
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Xiaodong Lü; Quanwu Mu. On Waring–Goldbach problem of mixed powers. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 523-538. doi : 10.5802/jtnb.951. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.951/

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