On Lagrange's four squares theorem with almost prime variables
type: Item
instance of: scholarly article
title: On Lagrange's four squares theorem with almost prime variables
zbMATH Open document ID: 1372.11096
author: Kai-Man Tsang
author: Lilu Zhao
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2017-05-03T00:00:00Z
full work available at URL: http://hdl.handle.net/10722/245057
review text: It is conjectured that every sufficiently large integer \(N\equiv 4\pmod{24}\) can be written as the sum of the squares of four primes, but we appear to be far from proving this. It was shown by the reviewer and \textit{D. I. Tolev} [J. Reine Angew. Math. 558, 159--224 (2003; Zbl 1022.11050)] that such \(N\) can be represented as \[ N=x_1^2+x_2^2+x_3^2+p^2 \] with \(p\) prime and each of \(x_1,x_2,x_3\) having at most 101 prime factors. This was improved by \textit{D. I. Tolev} [Bonn. Math. Schr. 360, 17 p. (2003; Zbl 1060.11061)] to 80, and by \textit{Y. Cai} [Int. J. Number Theory 6, No. 8, 1801--1817 (2010; Zbl 1219.11147)] to 42. The first result of the present paper is a further substantial improvement, showing that one can take each of \(x_1,x_2,x_3\) to have at most 5 prime factors. An alternative problem was considered by \textit{J. Brüdern} and \textit{E. Fouvry} [J. Reine Angew. Math. 454, 59--96 (1994; Zbl 0809.11060)], who showed that there are representations \[ N=x_1^2+x_2^2+x_3^2+x_4^2 \] in which each of \(x_1,x_2,x_3,x_4\) has at most 34 prime factors. The papers cited above, by the reviewer and Tolev, by Tolev, and by Cai, improved this successively, replacing 34 by 25, by 20, and by 13 respectively. The present paper makes a further significant improvement, showing that one may take each of \(x_1,x_2,x_3,x_4\) to have at most 4 prime factors. The proof of the first result follows the general structure of the paper by the reviewer and Tolev. However the authors apply Cauchy's inequality at a slightly different point, which turns out to be advantageous. Moreover, rather than use the vector sieve, Chen's reversal of rôles method is employed. For the second result the authors consider \[ N=x_1^2+x_2^2+x_3^2+q^2, \] with \(q\) almost-prime. Rather than use the vector sieve the paper combines a 3-dimensional sieve (for \(x_1,x_2\) and \(x_3\)) with the reversal of rôles method to handle \(q\).
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 6710714
zbMATH Keywords: Lagrange's theorem
zbMATH Keywords: almost-primes
zbMATH Keywords: primes
zbMATH Keywords: Chen's method
MaRDI profile type: Publication
OpenAlex ID: W2324835191
cites work: Lagrange's Four Squares Theorem with almost prime variables.
cites work: Q3152846
cites work: LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE
cites work: Q4081355
cites work: A New Application of the Hardy-Littlewood-Kloosterman Method
cites work: On the representation of a number in the form x^2+y^2+p^2+q^2 where p, q are odd primes
cites work: Q4052222
cites work: The square sieve and consecutive square-free numbers
cites work: Lagranges four squares theorem with one prime and three almost-prime variables
cites work: AN ASYMPTOTIC FORMULA FOR THE NUMBER OF SOLUTIONS OF A NONLINEAR EQUATION WITH PRIME NUMBERS
cites work: On the representation of the integer by positive quadratic forms with square-free variables
cites work: The Hardy-Littlewood method.
DOI: 10.1515/CRELLE-2014-0094
Recommended article: Lagrange's Four Squares Theorem with almost prime variables.
Recommended article: Lagranges four squares theorem with one prime and three almost-prime variables
Recommended article: On Lagrange's Theorem with Prime Variables
Recommended article: Some variants of Lagrange's four squares theorem
Recommended article: Refinements of Lagrange’s Four-Square Theorem
Recommended article: Q4831673
Recommended article: LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE
Recommended article: On the slim exceptional set for the Lagrange four squares theorem
Recommended article: Gauss's three squares theorem with almost prime variables
description: scientific article; zbMATH DE number 6710714