The ternary Goldbach problem with restrictions on the summands
type: Item
instance of: scholarly article
title: The ternary Goldbach problem with restrictions on the summands
zbMATH Open document ID: 0817.11045
DOI: 10.1007/BF02567619
published in: Manuscripta Mathematica
publication date: 1994-10-09T00:00:00Z
full work available at URL: https://eudml.org/doc/155966
review text: Using the combinatorial methods of \textit{P. Erdős} [Proc. Am. Math. Soc. 5, 847-853 (1954; Zbl 0056.270)], and \textit{G. Dufner} and \textit{D. Wolke} [Monatsh. Math. 118, 75-82 (1994; Zbl 0808.11060)] which exploit the fact that every large odd integer has many representations as a sum of three primes, \textit{E. Wirsing}'s result on thin bases of primes for the set of positive integers [Analysis 6, 285-308 (1986; Zbl 0586.10032)] is deduced from Vinogradov's three primes theorem and slightly generalized. The author proves that if \(\alpha_ j>0\) for \(j=1,2,3\), \(\alpha_ 1+ \alpha_ 2+ \alpha_ 3= 1\), and \(\beta_ 1+ \beta_ 2+ \beta_ 3=1\) then there exist subsets \(\mathbb{P}_ j\) of the set of all prime numbers \(\mathbb{P}\), such that \(| \mathbb{P}_ j\cap [1,x]| \ll x^{\alpha_ j} (\log x)^{\beta_ j}\), and every large odd number can be represented as \(p_ 1+ p_ 2+ p_ 3\), with \(p_ j\in \mathbb{P}_ j\). The author remarks that one can take \(\alpha_ 1=0\), provided that \(\beta_ 1\geq 2\). It is also possible to have two very thin sets of primes \(\mathbb{P}_ 1\) and \(\mathbb{P}_ 2\) (with \(\alpha_ 1= \alpha_ 2 =0\) and \(\beta_ 1= \beta_ 2 =1\)), at the cost of taking \(\mathbb{P}_ 3= \mathbb{P}\).
reviewed by: Alessandro Zaccagnini
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 624941
zbMATH Keywords: ternary Goldbach problem
zbMATH Keywords: additive bases
zbMATH Keywords: representation of integers
zbMATH Keywords: combinatorial methods
zbMATH Keywords: Vinogradov's three primes theorem
author: Dieter Wolke
MaRDI profile type: Publication
cites work: Some Results on Additive Number Theory
cites work: THIN SUBBASES
cites work: On the representation of large odd integer as a sum of three almost equal primes
description: scientific article; zbMATH DE number 624941