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## Colloquium Mathematicum

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## A remark on Zolotarev’s theorem

### Volume 171 / 2023

Colloquium Mathematicum 171 (2023), 159-166 MSC: Primary 05A05; Secondary 11A07, 11A15. DOI: 10.4064/cm8738-7-2022 Published online: 5 September 2022

#### Abstract

For an odd integer $n \ge 3$ and $a\in \mathbb {Z}$ with $(a,n)=1$, let $$\mathcal {U}_{a,n}=\biggl \{1\le x\le \frac {n-1}{2}: \{ax\}_n\ge \frac {n+1}{2}\bigg \},$$ $$\mathcal {I}_{a,n}=\biggl \{(i,j): 1\le i \lt j\le \frac {n-1}{2}\text { and }\|ai\|_n \gt \|aj\|_n\bigg \},$$ where $\{x\}_n$ denotes the least non-negative residue of $x$ modulo $n$ and $$\|x\|_n:=\min \,\{\{x\}_n,n-\{x\}_n\}.$$ We show that $$|\mathcal {I}_{a,n}|= |\mathcal {U}_{a,n}|\cdot \biggl (\frac {n-1}{2}-|\mathcal {U}_{a,n}| \bigg ).$$ As a consequence, we obtain $$\mathop {\rm sign}\gamma _{a,n}=\begin {cases} \bigl (\frac {a}{n}\big )&\text {if }n\equiv 1\pmod {4},\\ 1&\text {if }n\equiv 3\pmod {4}, \end {cases}$$ where $\bigl(\frac{\cdot }{\cdot }\bigr)$ denotes the Jacobi symbol and $\gamma _{a,n}$ is the permutation of $\{1,\ldots ,(n-1)/2\}$ defined by $\gamma _{a,n}(x)=\|ax\|_n$.

#### Authors

• Chao HuangSchool of Mathematics and Physics
Anqing Normal University
Anqing 246133
People’s Republic of China
e-mail
• Hao PanSchool of Applied Mathematics
Nanjing University of Finance and Economics
Nanjing 210046
People’s Republic of China
e-mail

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