Power Residues and Nonresidues in Arithmetic Progressions

Let A: be an integer > 2 andp a prime such that vk(p) = (k,p — 1) > 1. Let bn + c(n = 0,1,. ..;b > 2,1 < 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 r2(p,b,c), and the nth smallest prime kth power nonresidue by g„(p,k), n = 0, 1, 2,_ If C(p) is the multiplicative group consisting of the residue classes mod p, then the fcth powers mod p form a multiplicative subgroup, Ck(p). Among the vk(p) cosets of Ck(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 Tso 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 2V*b5/2p2/5 + 3b3pv> + b2 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\(p,k))ls. We further obtain a similar bound for r2(p,b,c) for every prime p, without exception, and we apply our results to obtain a bound of the order of p2/5 for the nth smallest prime A:th power nonresidue of primes which are large relative to UJ"—1 gj(p>k)