Prime power residue and linear coverings of vector space over \(\mathbb{F}_q\) (Q6156909)

Let \(q\) be an odd prime. The paper studies necessary and sufficient conditions under which a finite set \(B\) of nonzero integers that does not contain a perfect \(q\) power contains a \(q\) power modulo \(p\) for all but finitely many primes \(p\). The main result is that the set \(B\) has the above property if and only if \(B\) corresponds to a linear hyperplane covering of \({\mathbb F}_q^k\). Here, \(k\) is the number of prime factors involved in the \(q\)-free parts of the elements of \(B\). Consequently, a set \(B\subset {\mathbb Z}\backslash \{0\}\) of cardinality less that \(q+1\) cannot contain a \(q\)th power modulo \(p\) for almost all primes \(p\) unless it already contains a perfect \(q\) power. Furthermore, for every set of \(l\) nonzero integers \(B=\{b_j, 1\le j\le l\}\) and vector of \(l\) nonzero exponents modulo \(q\), \(\mathbf{c}:=(c_j)_{j=1}^l\in ({\mathbb Z}_q\backslash \{0\})^l\), the set \(B\) has the above property for almost all primes \(p\) if and only if the set \(B^{\mathbf c}:=\{b_j^{c_j}, 1\le j\le l\}\) has the same property. The proofs use the Chebotarev density theorem and linear algebra.