On the quadratic residues and their distribution properties (Q6083285)

For an odd prime \(p\) denote by \(\bar a\) the solution of the congruence \(ax\equiv 1\bmod p\), and let \(N(p,1)\) resp. \(N(p,-1)\) be the number of quadratic residues, resp. non-residues \(a\bmod p\) such that \(a\pm\bar a\) are also quadratic residues mod \(p\). Recently \textit{T. T. Wang} and \textit{X. X. Lv} [Symmetry 12, No. 3, 241 (2020; \url{doi:10.3390/sym12030421})] gave an explicit formula for \(N(p,\pm1)\) in the case \(p\equiv3\bmod 4\) and an asymptotic formula for \(p\equiv1\bmod 4\). Now the authors provide an explicit formula in the case \(p\equiv1\bmod 4\).