论文标题
$ \ {1,2，\ ldots，（p-1）/2 \} $ for $ p \ equiv 1 \ pmod {4} $均衡的削皮
Balanced paring of $\{1,2,\ldots,(p-1)/2\}$ for $p\equiv 1 \pmod{4}$
论文作者
论文摘要
令$ p \ equiv 1 \ pmod {4} $为prime。写$ t = \ prod_ {x = 1}^{（p-1）/2} x $。由于$ t ^2 \ equiv -1 \ pmod {p} $，我们可以划分$ \ {1,2，\ ldots，（p-1）/2 \} $成$（p-1）/4 $订购对，以便每对，例如$ <a，\ tilde {a}> $，满足$ tilde {a}> pm pm pm pm pm pm pm \ e \ pm \ pm \ pm { \ pmod {p}。$对于任何两个这样的对，假设$ a <\ tilde {a}，b <\ tilde {b}，a <b $，那么它们的相对顺序有三个可能性：$ a <\ tilde {a} a} <b <b <b <\ tilde {b} $ a <b <b <b <b <b <b <b <b <b < ，$ a <b <\ tilde {b} <\ tilde {a} $。我们表明，在三种情况下以相等的频率出现的意义上，这种削皮是平衡的。利用这种削皮的特性，我们解决了Zhi-wei Sun提出的一个问题，该问题与二次残基有关的排列符号。
Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$. Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into $(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that $t a \equiv \pm \tilde{a} \pmod{p}.$ 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.
