query: Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$?
positive_passages.docid: 19761
positive_passages.text: Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime.
negative_passages.docid: 81875
negative_passages.text: why is $\sqrt{-1} = i$ and not $\pm i$?
negative_passages.docid: 93756
negative_passages.text: $1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$
negative_passages.docid: 26478
negative_passages.text: Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
negative_passages.docid: 77470
negative_passages.text: Find modulus of $\frac{|z_1-z_2|}{|1-(z_1)(\overline{z_2})|}$
negative_passages.docid: 15164
negative_passages.text: Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
negative_passages.docid: 104484
negative_passages.text: Prove that $r$ is a primitive root modulo $p$ if and only if $r^{(p−1)/q}\not\equiv 1\pmod{p}$
negative_passages.docid: 134348
negative_passages.text: When is $(p - 1)! + 1$ a power of $p$?
negative_passages.docid: 96039
negative_passages.text: Why is $\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}\le \frac 1 {n+1}$
negative_passages.docid: 121052
negative_passages.text: Where is the mistake? $-1=(-1)^{2/2}=\left(\left (-1\right)\displaystyle ^2\right)^{1/2}=1^{1/2}=\sqrt 1=1$
negative_passages.docid: 52461
negative_passages.text: Why does $(g^b \bmod m)^a \bmod m = (g^a \bmod m)^b \bmod m$?
negative_passages.docid: 35478
negative_passages.text: Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
negative_passages.docid: 117816
negative_passages.text: Is $\frac{x^2+x}{x+1}$ a polynomial?
negative_passages.docid: 140825
negative_passages.text: $\sum_{n=1}^{p-1}{\frac{1}{n}} = \frac{A_p}{B_p}$ What is $A_p$ (mod $p^2$) where $\frac{A_p}{B_p}$ is a reduced form fraction?
negative_passages.docid: 51864
negative_passages.text: $\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
negative_passages.docid: 40534
negative_passages.text: What is $(-1)^{\frac{2}{3}}$?
negative_passages.docid: 17707
negative_passages.text: Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?