A conjecture of Sárközy on quadratic residues (Q1981584)

The paper addresses a conjecture of \textit{A. Sárközy} [Acta Arith. 155, No. 1, 41--51 (2012; Zbl 1357.11100)] on the indecomposability of the quadratic residues. For a prime \(p\) let \(R_p\) denote the set of all quadratic residues modulo \(p\). Sárközy conjectured that \(R_p\) has no 2-decomposition \(R_p=A+B\) with \(|A|,|B|\geq 2\). Sárközy also proved that if \(p\) is a sufficiently large prime, then \(R_p\) has no 3-decomposition \(R_p=A+B+C\) with \(|A|,|B|,|C|\geq 2\). In this paper the authors extend this result to \textit{all} primes. Furthermore, they also prove that for any prime \(p\), if \(U+V=R_p\) is a 2-decomposition with \(|U|,|V|\geq 2\), then \[ \frac{7-\sqrt{17}}{16}\sqrt{p}+1\leq |U|,|V|\leq \frac{7+\sqrt{17}}{4}-6.63,\] improving on results of Sárközy, Shparlinski, Shkredov of this type.