Balanced paring of $\{1,2,\dots ,(p-1)/2\}$ for $p\equiv 1(\mathrm{mod}4)$

Abstract

Let $p\equiv 1(\mathrm{mod}4)$ be a prime. Write $t={\prod }_{x=1}^{(p-1)/2}x$. Since $t^2\equiv -1(\mathrm{mod}p)$ , we can divide $\{1,2,\dots ,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $ta\equiv \pm \tilde{a}(\mathrm{mod}p).$ For any two such pairs, assume $a<\tilde{a},b<\tilde{b},a<b$, then there are three possibilities for their relative order : $a<\tilde{a}<b<\tilde{b}$ , $a<b<\tilde{a}<\tilde{b}$ , $a<b<\tilde{b}<\tilde{a}$. We show this paring is balanced in the sense that the three cases occur with equal frequencies. Utilizing properties of this paring we solve one problem raised by Zhi-Wei Sun concerning the sign of permutation related to quadratic residues.

References in corpus (1)

• A remark on Zoloterav's theorem