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APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES

Part of: Combinatorics Sequences and sets Elementary number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  10 July 2019

LI-YUAN WANG Open the ORCID record for LI-YUAN WANG [Opens in a new window]  and
HAI-LIANG WU Open the ORCID record for HAI-LIANG WU [Opens in a new window]
LI-YUAN WANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China email [email protected]
HAI-LIANG WU*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

Keywords

Zolotarev’s lemmapermutationquadratic residueLerch’s theoremprimitive root

MSC classification

Primary: 11A15: Power residues, reciprocity
Secondary: 05A05: Permutations, words, matrices 11A07: Congruences; primitive roots; residue systems 11B75: Other combinatorial number theory 11R11: Quadratic extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 362 - 371
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

This research was supported by the National Natural Science Foundation of China (grant no. 11571162).

References

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