Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913943554883584 |
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| author | Mishra, Bhawesh Santonastaso, Paolo |
| author_facet | Mishra, Bhawesh Santonastaso, Paolo |
| contents | 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 $\PG(\mathbb{F}_{q}^{n})$. 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 $\mathrm{PGL}(n, q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors Mishra, Bhawesh Santonastaso, Paolo Number Theory 11A15, 51E21 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 $\PG(\mathbb{F}_{q}^{n})$. 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 $\mathrm{PGL}(n, q)$. |
| title | Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors |
| topic | Number Theory 11A15, 51E21 |
| url | https://arxiv.org/abs/2507.11828 |