Representation of even integers as sums of squares of primes and powers of 2
type: Item
instance of: scholarly article
title: Representation of even integers as sums of squares of primes and powers of 2
zbMATH Open document ID: 0961.11035
author: Ming-Chit Liu
author: Jianya Liu
published in: Journal of Number Theory
publication date: 2001-06-05T00:00:00Z
review text: It has been shown by \textit{J. Liu}, \textit{M.-C. Liu} and \textit{T. Zhan} [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)] that there is an integer \(k\) such that every sufficiently large even number may be written in the form \[ p^2_1+p^2_2+ p^2_3+p^2_4+ 2^{e_1}+ \cdots+ 2^{e_k}. \] Here it is shown that one may be take \(k=8330\). There are a couple of significant departures from the work of Liu, Liu and Zhan. Most notably, one has to estimate the number of solutions of the equation \[ n=p^2_1+ p^2_2- p^2_3- p^2_4 \] for a given non-zero integer \(n\), and the present paper uses the `vector sieve' of \textit{J. Brüdern} and \textit{E. Fouvry} [J. Reine Angew. Math. 454, 59-96 (1994; Zbl 0809.11060)]. Although this is a clear improvement, a more direct approach is clearly desirable.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 1515455
zbMATH Keywords: representation of even integers
zbMATH Keywords: sums of squares of primes and powers of 2
zbMATH Keywords: vector sieve
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
OpenAlex ID: W1964406663
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DOI: 10.1006/JNTH.1999.2500
description: scientific article; zbMATH DE number 1515455