Prime power residue and linear coverings of vector space over [formula omitted

Abstract

Let q be an odd prime and B={bj}j=1l be a finite set of nonzero integers that does not contain a perfect qth power. We show that B has a qth power modulo every prime p≠q and not dividing ∏b∈Bb if and only if B corresponds to a linear hyperplane covering of Fqk. Here, k is the number of distinct prime factors of the q-free part of elements of B. Consequently: (i) a set B⊂Z∖{0} with cardinality less than q+1 cannot have a qth power modulo almost every prime unless it contains a perfect qth power and (ii) For every set B={bj}j=1l⊂Z∖{0} and for every (cj)j=1l∈(Fq∖{0})l the set B contains a qth power modulo every prime p≠q and not dividing ∏j=1l if and only if the set {bjcj}j=1l does so.