Power residues and nonresidues in arithmetic progressions

Abstract:

Let k be an integer $\geq 2$ and p a prime such that ${v_k}(p) = (k,p - 1) > 1$. Let $bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c < b,(b,p) = (c,p) = 1)$ be an arithmetic progression. We denote the smallest kth power nonresidue in the progression $bn + c$ by $g(p,k,b,c)$, the smallest quadratic residue in the progression $bn + c$ by ${r_2}(p,b,c)$, and the nth smallest prime kth power nonresidue by ${g_n}(p,k),n = 0,1,2, \ldots$. If $C(p)$ is the multiplicative group consisting of the residue classes $\bmod \;p$, then the kth powers $\bmod \;p$ form a multiplicative subgroup, ${C_k}(p)$. Among the ${v_k}(p)$ cosets of ${C_k}(p)$ denote by T the coset to which c belongs (where c is the first term in the progression $bn + c)$, and let $h(p,k,b,c)$ denote the smallest number in the progression $bn + c$ which does not belong to T so that $h(p,k,b,c)$ is a natural generalization of $g(p,k,b,c)$. We prove by purely elementary methods that $h(p,k,b,c)$ is bounded above by ${2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}$ if p is a prime for which either b or $p - 1$ is a kth power nonresidue. The restriction on b and $p - 1$ may be lifted if $p > {({g_1}(p,k))^{7.5}}$. We further obtain a similar bound for ${r_2}(p,b,c)$ for every prime p, without exception, and we apply our results to obtain a bound of the order of ${p^{2/5}}$ for the nth smallest prime kth power nonresidue of primes which are large relative to $\Pi _{j = 1}^{n - 1}{g_j}(p,k)$.