On the least quadratic nonresidues (mod \(p\)) (Q687922)

For an odd prime \(p\) let \(\alpha(p)\) denote the smallest positive integer \(n\) such that \(({n\over p})= ({{n+1} \over p})=-1\), where \(({n\over p})\) is Legendre's symbol. The best known evaluation \[ \alpha(p)\ll_ \varepsilon p^{{1\over 4}(1-{1\over 2}e^{-10})+\varepsilon}, \qquad \varepsilon>0, \] was obtained by \textit{P. D. T. A. Elliott} [Probabilistic number theory (1979; Zbl 0431.10029), 147-158]. In the present paper the author modifies Elliott's method and proves \[ \alpha(p)\ll_ \varepsilon p^{{1\over 4}(1- {1\over 2} e^{-9})+\varepsilon}, \qquad \varepsilon>0. \]