Small Prime kth Power Residues and Nonresidues in Arithmetic Progressions

Let p be a large odd prime, let x=(log p)1+ε and let qloglog p be an integer, where ε>0 is a small number. This note proves the existence of small prime quadratic residues and prime quadratic nonresidues in the arithmetic progression a+qm x, with relatively prime 1≤ a<q, unconditionally. The same results are generalized to small prime kth power residues and nonresidues, where k p-1 and kloglog p.