Open Access
June 2015 Counting additive decompositions of quadratic residues in finite fields
Simon R. Blackburn, Sergei V. Konyagin, Igor E. Shparlinski
Funct. Approx. Comment. Math. 52(2): 223-227 (June 2015). DOI: 10.7169/facm/2015.52.2.3


We say that a set $\mathcal{S}$ is additively decomposed into two sets $\mathcal{A}$ and $\mathcal{B}$ if $\mathcal{S} = \{a+b: a\in \mathcal{A}, b \in \mathcal{B}\}$. A. Sárközy has recently conjectured that the set $\mathcal{Q}$ of quadratic residues modulo a prime $p$ does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions.


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Simon R. Blackburn. Sergei V. Konyagin. Igor E. Shparlinski. "Counting additive decompositions of quadratic residues in finite fields." Funct. Approx. Comment. Math. 52 (2) 223 - 227, June 2015.


Published: June 2015
First available in Project Euclid: 18 June 2015

zbMATH: 06862259
MathSciNet: MR3358317
Digital Object Identifier: 10.7169/facm/2015.52.2.3

Primary: 11B13
Secondary: 11L40

Keywords: additive decompositions , finite fields , quadratic nonresidues character sums

Rights: Copyright © 2015 Adam Mickiewicz University

Vol.52 • No. 2 • June 2015
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