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A NEW CHARACTERISATION FOR QUARTIC RESIDUACITY OF $\mathbf {2}$

Part of: Combinatorics Elementary number theory

Published online by Cambridge University Press:  14 March 2022

CHAO HUANG Open the ORCID record for CHAO HUANG [Opens in a new window]  and
HAO PAN Open the ORCID record for HAO PAN [Opens in a new window]
CHAO HUANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China e-mail: DG1921004@smail.nju.edu.cn
HAO PAN*
Affiliation:
School of Applied Mathematics, Nanjing University of Finance and Economics Nanjing 210023, PR China
*
e-mail: haopan79@zoho.com
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Abstract

Let p be a prime with $p\equiv 1\pmod {4}$. Gauss first proved that $2$ is a quartic residue modulo p if and only if $p=x^2+64y^2$ for some $x,y\in \Bbb Z$ and various expressions for the quartic residue symbol $(\frac {2}{p})_4$ are known. We give a new characterisation via a permutation, the sign of which is determined by $(\frac {2}{p})_4$. The permutation is induced by the rule $x \mapsto y-x$ on the $(p-1)/4$ solutions $(x,y)$ to $x^2+y^2\equiv 0 \pmod {p}$ satisfying $1\leq x < y \leq (p-1)/2$.

Keywords

quartic residuepermutationclass number

MSC classification

Primary: 11A15: Power residues, reciprocity
Secondary: 05A05: Permutations, words, matrices 11A07: Congruences; primitive roots; residue systems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 1 - 6
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author is supported by the National Natural Science Foundation of China (Grant No.11971222). The second author is supported by the National Natural Science Foundation of China (Grant No.12071208).

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