Two results on powers of 2 in Waring-Goldbach problem
type: Item
instance of: scholarly article
title: Two results on powers of 2 in Waring-Goldbach problem
zbMATH Open document ID: 1273.11148
author: Zhixin Liu
author: Guangshi Lü
published in: Journal of Number Theory
publication date: 2011-03-03T00:00:00Z
review text: Following the line of investigation of \textit{Yu. V. Linnik} [Tr. Mat. Inst. Steklova 38, 152--169 (1951; Zbl 0049.31402)] that every large even integer is the sum of two odd primes and a fixed number of powers of two, many authors have improved Linnik's result and also generalized it to similar representations with higher powers of primes. In the present paper the authors prove the following two results: (1) every large odd integer is the sum of one prime, four cubes of primes and 106 powers of two, (2) every large even integer is the sum of two squares of primes, four cubes of primes and 211 powers of two. Furthermore, under the conjecture of an asymptotic formula predicted by Hardy-Littlewood on the number of representations of non-zero number as the difference of two primes, the 106 in (1) can be improved to 74. The method of proof uses the circle method. On the minor arcs, exponential sums over squares and cubes of primes are estimated by recent results of \textit{A. V. Kumchev} [Mich. Math. J. 54, No. 2, 243--268 (2006; Zbl 1137.11054)].
reviewed by: Kai-Man Tsang
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 5859976
zbMATH Keywords: Waring-Goldback problem
zbMATH Keywords: Powers of 2
zbMATH Keywords: Circle method
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.11.007
OpenAlex ID: W2059071826
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DOI: 10.1016/J.JNT.2010.11.007
description: scientific article; zbMATH DE number 5859976