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APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 3 / December 2019
- Published online by Cambridge University Press:
- 10 July 2019, pp. 362-371
- Print publication:
- December 2019
-
- Article
- Export citation
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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 
$p$.
