Quadratic residues and related permutations
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866909518088110080 |
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| author | Wu, Hai-Liang |
| author_facet | Wu, Hai-Liang |
| contents | Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\\ &A_2: \overline{g^2},\ \overline{g^4},\ \cdots,\ \overline{g^{p-1}}, \end{align*} where $g\in\Z$ is a primitive root modulo $p$ and $1\le a_1<a_2<\cdots<a_{(p-1)/2}\le p-1$ are all quadratic residues modulo $p$. Obviously $A_i$ is a permutation of $A_j$ and we call this permutation $σ_{i,j}$. Sun obtained the sign of $σ_{0,1}$ when $p\equiv 3\pmod4$. In this paper we give the sign of $σ_{0,1}$ and determine the sign $σ_{0,2}$ when $p\equiv 1\pmod 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1903_01098 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Quadratic residues and related permutations Wu, Hai-Liang Number Theory Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\\ &A_2: \overline{g^2},\ \overline{g^4},\ \cdots,\ \overline{g^{p-1}}, \end{align*} where $g\in\Z$ is a primitive root modulo $p$ and $1\le a_1<a_2<\cdots<a_{(p-1)/2}\le p-1$ are all quadratic residues modulo $p$. Obviously $A_i$ is a permutation of $A_j$ and we call this permutation $σ_{i,j}$. Sun obtained the sign of $σ_{0,1}$ when $p\equiv 3\pmod4$. In this paper we give the sign of $σ_{0,1}$ and determine the sign $σ_{0,2}$ when $p\equiv 1\pmod 4$. |
| title | Quadratic residues and related permutations |
| topic | Number Theory |
| url | https://arxiv.org/abs/1903.01098 |