Prime Power Residues and Blocking Sets

TL;DR

This paper establishes a connection between prime power residues and blocking sets in projective geometry, providing a classification and bounds for minimal sets that contain q-th powers modulo almost all primes.

Contribution

It introduces a novel link between number theory and Galois geometry, characterizing sets with prime power residue properties via geometric blocking sets.

Findings

Sets with prime power residue properties correspond to blocking sets in projective geometry.

The property of containing q-th powers modulo almost all primes is invariant under geometric q-equivalence.

The paper classifies and bounds the sizes of minimal such sets.

Abstract

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to hyperplanes) in $\mathrm{PG}(\mathbb{F}_{q}^{k})$. 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 $q^{th}$ power modulo almost every prime $p$ is invariant under geometric $q$-equivalence defined by an element of the projective general linear group $\mathrm{PGL}(\mathbb{F}_{q}^{k})$. 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 $B$.