On quadratic residues and a conjecture of Sárközy.

A typical problem in additive combinatorics is to find the structure of a set given an additive assumption about the set. Sárközy's conjecture is one such problem. The conjecture posits the nonexistence of a nontrivial 2-decomposition of the set Q_p of quadratic residue modulo p. In this paper, we review the history and formulation of the conjecture. Then we survey the progress made towards this conjecture in which we first suppose that a 2-decomposition Q_p = A+B exists, with │A│, │B│ ≥2. Next, we find bounds for the cardinalities of sets A and B while showcasing the techniques used in obtaining them. Finally, we suggest further problems by discussing the case for a 3-decomposition of Q_p and other related problems. [ABSTRACT FROM AUTHOR]