Prime Power Residues and Blocking Sets

Let q be a fixed odd prime. We show that a finite subset B of integers, not containing any perfect q t h power, contains a q t h power modulo almost every prime if and only if B corresponds to a blocking set (with respect to hyperplanes) in PG ( 𝔽 q k ) . Here, k is the number of distinct prime divisors of q -free parts of elements of B . As a consequence, the property of a subset B to contain q t h power modulo almost every prime p is invariant under geometric q -equivalence defined by an element of the projective general linear group PGL ( 𝔽 q k ) . Employing this connection between two disparate branches of mathematics, Galois geometry and number theory, we classify, and provide bounds on the sizes of, minimal such sets B .