Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

Let $p$ be a large odd prime, let $x=\log p)(\log \log p{)}^{3+\epsilon }$ and let $q\ll \log \log p$ be an integer, where $\epsilon >0$ is a small number. This note proves the existence of small prime quadratic residues and small prime quadratic nonresidues in the arithmetic progression $a+qm\ll x$, with relatively prime $1\le a<q$, unconditionally. The same results are generalized to small prime $k$th power residues and nonresidues, where $k\mid p-1$ and $k\ll \log \log p$.