2018/10/06 by Li-Yuan Wang, Hai-Liang Wu, Wang, Li-Yuan +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1810.03006
11 pages
arxiv created 2019/02/08 · arxiv updated 2019/02/11
Let p be an odd prime. For each integer a with p\nmid a, the famous Zolotarev's Lemma says that the Legendre symbol ((a)/(p)) is the sign of the permutation of \Z/p\Z induced by multiplication by a. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary k-th power residue modulo p and primitive roots of a power of p. Finally, we discuss some permutation problems concerning quadratic residues modulo p. In particular, we confirm some conjectures posed by Sun.