Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors

2025/07/16 by Bhawesh Mishra, Paolo Santonastaso, Mishra, Bhawesh +1
Computer Science · Mathematics · #11A15 #51E21 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.11828

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

Let q be an odd prime and k be a natural number. We show that a finite subset of integers S that does not contain any perfect q^th power, contains a q^th power residue modulo almost every natural numbers N with at most k prime factors if and only if S corresponds to a k-blocking set of \PG(\mathbbF_qⁿ). Here, n is the number of distinct primes that divides the q-free parts of elements of S. Consequently, this geometric connection enables us to utilize methods from Galois geometry to derive lower bounds for the cardinalities of such sets S and to completely characterize such S of the smallest and the second smallest cardinalities. Furthermore, the property of whether a finite subset of integers contains a q^th power residue modulo almost every integer N with at most k prime factors is invariant under the action of projective general linear group PGL(n, q).

Citations

• Polynomials with factors of the form (x^q-a) with roots modulo every integer

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