Sumsets being squares
type: Item
instance of: scholarly article
title: Sumsets being squares
zbMATH Open document ID: 1313.11120
author: Andrej Dujella
author: Christian Elsholtz
published in: Acta Mathematica Hungarica
publication date: 2015-01-09T00:00:00Z
review text: An old problem of Euler on sumsets \(A + B\) asks for which \(A\) and \(B\) all elements of \(A + B\) are integer squares. Note that if \(| A| = 2\), then the size of \(| B| \) is bounded by a divisor function. On the other hand, if \(| A| \geq 4\) \textit{N. Alon} et al. [Isr. J. Math. 188, 353--384 (2012; Zbl 1288.05124)] observed, assuming a deep Bombieri-Lang conjecture on curves of genus \(g > 1\), that the size of \(| B| \) is uniformly bounded (cf. also \textit{J. Solymosi} [Additive combinatorics, CRM Proc. Lect. Notes 43, 29--38 (2007; Zbl 1133.05096)]). The authors prove the following results: (1) There exists a set \(A\) of 3 positive integers and a corresponding set \(B \subset [0,N]\) with \(| B| \gg (\log N)^{15/17}\), such that all elements of \(A+B\) are perfect squares. (2) There exists a set \(A\) of 3 integers and a corresponding set \(B \subset[0,N]\) with \(| B| \gg (\log N)^{9/11}\), such that all elements of the sets \(A\), \(B\) and \(A + B\) are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank. The first result improves the exponent \(5/7\) proved by \textit{N. Alon} et al. (loc. cit.)] for the case \(B\subset [-N,N]\).
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6385259
zbMATH Keywords: sumset
zbMATH Keywords: square
zbMATH Keywords: elliptic curve of high rank
Wikidata QID: Q57595847
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: s10474-013-0334-8
OpenAlex ID: W2165773351
cites work: Sums and products along sparse graphs
cites work: On generalizations of a problem of Diophantus
cites work: Subset sums avoiding quadratic nonresidues
cites work: Q4341641
cites work: Hilbert cubes in progression-free sets and in the set of squares
cites work: On Mordell-Weil groups of elliptic curves induced by Diophantine triples
cites work: Diophantine triples and construction of high-rank elliptic curves over \(\mathbb{Q}\) with three nontrivial 2-torsion points
cites work: High-rank elliptic curves with torsion induced by Diophantine triples
cites work: The factor-difference set of integers
cites work: On a problem of Diophantus
cites work: On sums which are powers
cites work: A note on a conjecture of Erdős and Rosenfeld
cites work: Congruence properties of the \(\Omega\)-function on sumsets
cites work: On additive decompositions of the set of quadratic residues modulo \(p\)
cites work: Q5431581
DOI: 10.1007/S10474-013-0334-8
description: scientific article; zbMATH DE number 6385259