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APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES

  • LI-YUAN WANG (a1) and HAI-LIANG WU (a2)

Abstract

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  $p$ .

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

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[1] Brunyate, A. and Clark, P. L., ‘Extending the Zolotarev–Frobenius approach to quadratic reciprocity’, Ramanujan J. 37 (2015), 2550.
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[8] Szekely, G. J. (ed), Contests in Higher Mathematics (Springer, New York, 1996).
[9] Zolotarev, G., ‘Nouvelle démonstration de la loi de réciprocité de Legendre’, Nouvelles Ann. Math. 11 (1872), 354362.
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