Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916665264963584 |
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| author | Mishra, Bhawesh |
| author_facet | Mishra, Bhawesh |
| contents | Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pm p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{N}^{e_{N}} : 0 \leq e_{1}, e_{2}, \ldots, e_{N}\leq N\}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $m_{q,k} \frac{N^{Nk}}{q^{N}}$ for some positive constant $m_{q,k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13167 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime Mishra, Bhawesh Number Theory 11A15, 11A07 Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pm p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{N}^{e_{N}} : 0 \leq e_{1}, e_{2}, \ldots, e_{N}\leq N\}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $m_{q,k} \frac{N^{Nk}}{q^{N}}$ for some positive constant $m_{q,k}$. |
| title | Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime |
| topic | Number Theory 11A15, 11A07 |
| url | https://arxiv.org/abs/2308.13167 |