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 $B$.
Soit $q$ un nombre premier impair et soit $B$ une partie finie de l’ensemble des entiers qui ne contient aucune puissance $q$-ième parfaite. Nous prouvons que $B$ contient une puissance $q$-ième modulo presque tout nombre premier si et seulement si $B$ correspond à un ensemble bloquant (par rapport aux hyperplans) dans $\mathrm{PG}({\mathbb{F}}_{q}^{k})$. Ici $k$ désigne le nombre de diviseurs premiers des parties $q$-libres des éléments de $B.$ Par conséquent, la propriété de contenir une puissance $q$-ième modulo presque tout nombre premier est invariante sous la $q$-équivalence géométrique définie par un élément du groupe projectif linéaire $\mathrm{PGL}(\mathbb{F}_{q}^{k})$. En utilisant ce lien entre la géométrie de Galois et la théorie des nombres, deux branches différentes des mathématiques, nous classifions les ensembles minimaux vérifiant cette propriété et donnons des majorations de leur taille.
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Keywords: Power-Residue, Blocking Sets, Local-Global Principle
Bhawesh Mishra 1 ; Paolo Santonastaso 2
CC-BY-ND 4.0
@article{JTNB_2025__37_3_775_0,
author = {Bhawesh Mishra and Paolo Santonastaso},
title = {Prime {Power} {Residues} and {Blocking} {Sets}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {775--793},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {3},
doi = {10.5802/jtnb.1342},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1342/}
}
TY - JOUR AU - Bhawesh Mishra AU - Paolo Santonastaso TI - Prime Power Residues and Blocking Sets JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 775 EP - 793 VL - 37 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1342/ DO - 10.5802/jtnb.1342 LA - en ID - JTNB_2025__37_3_775_0 ER -
%0 Journal Article %A Bhawesh Mishra %A Paolo Santonastaso %T Prime Power Residues and Blocking Sets %J Journal de théorie des nombres de Bordeaux %D 2025 %P 775-793 %V 37 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1342/ %R 10.5802/jtnb.1342 %G en %F JTNB_2025__37_3_775_0
Bhawesh Mishra; Paolo Santonastaso. Prime Power Residues and Blocking Sets. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 775-793. doi: 10.5802/jtnb.1342
[1] Un’estensione del teorema di Segre-Kustaanheimo, Boll. Unione Mat. Ital., Volume 10 (1955) no. 4, pp. 498-506 | Zbl | MR
[2] On the size of a blocking set in PG, Combinatorica, Volume 14 (1994), pp. 111-114 | MR | DOI | Zbl
[3] Blocking sets in projective spaces, Current research topics in Galois geometry (Mathematics Research Developments), Nova Science Publishers/Novinka, 2011, pp. 61-84 | Zbl
[4] A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, J. Comb. Theory, Volume 1 (1966) no. 1, pp. 96-104 | Zbl | DOI | MR
[5] Hyperplane coverings and blocking sets, Math. Z., Volume 181 (1982) no. 3, pp. 407-409 | DOI | Zbl | MR
[6] Classification of minimal blocking sets in small Desarguesian projective planes, J. Comb. Des., Volume 30 (2022) no. 8, pp. 561-580 | Zbl | DOI | MR
[7] Sets which contain a quadratic residue modulo for almost all , Math. J. Okayama Univ., Volume 39 (1989), pp. 1-8 | Zbl | MR
[8] Arithmetical properties of value sets of polynomials, Acta Arith., Volume 15 (1969) no. 2, pp. 91-115 | DOI | Zbl | MR
[9] Blockierende Mengen in endlichen projektiven Räumen, Ph. D. Thesis, Justus-Liebig-Universität Giessen, Giessen (1996)
[10] Projective geometries over finite fields, Oxford Mathematical Monographs, Clarendon Press; Oxford University Press, 1998 | DOI | Zbl | MR
[11] Prime power residue and linear coverings of vector space over , Finite Fields Appl., Volume 89 (2023), 102199, 12 pages | Zbl | MR
[12] Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito., Boll. Unione Mat. Ital., Volume 10 (1955) no. 4, pp. 507-513 | Zbl | MR
[13] On power residues, Acta Arith., Volume 108 (2003) no. 1, pp. 77-94 | DOI | MR | Zbl
[14] On sets which contain a power residue for almost all prime modules, Colloq. Math., Volume 102 (2005), pp. 67-71 | MR | DOI | Zbl
[15] On large minimal blocking sets in PG(2, q), J. Comb. Des., Volume 13 (2005) no. 1, pp. 25-41 | DOI | MR | Zbl
[16] Blocking sets with respect to planes in and maximal spreads of a non-singular quadric in , Mitt. Math. Semin. Gießen, Volume 201 (1991), pp. 141-147 | MR | Zbl
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