Prime Power Residues and Blocking Sets
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911049318400000 |
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| author | Mishra, Bhawesh Santonastaso, Paolo |
| author_facet | Mishra, Bhawesh Santonastaso, Paolo |
| contents | 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 $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime Power Residues and Blocking Sets Mishra, Bhawesh Santonastaso, Paolo Number Theory Combinatorics 51E21, 05B25, 11A15 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 $B$. |
| title | Prime Power Residues and Blocking Sets |
| topic | Number Theory Combinatorics 51E21, 05B25, 11A15 |
| url | https://arxiv.org/abs/2507.07673 |