Let $n$ be a positive integer and $p=4n+1$, a prime number. Show that $(\frac{p-1}2)!$ is a root of $x^2+1=0$ in $\mathbb Z_p$
119630
For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
34430
Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$?
156607
Show that $L_p =\frac{1^p+2^p+3^p+...+(p-1)^p}{p}$ is an integer and generalize the integer for all odd integers p
45645
Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
154475
Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
147101
Show that if $p>5$ is prime, then $240 \mid (p^4-1)$
43940
Showing that $n$ is pseudoprime to the base $a$
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
75528
$x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
39915
Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
90043
Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$.
5978
Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
30374
If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
145509
Prove the roots of $p(z)+z^n\bar{p}(\frac{1}{z})$ lie on the unit circle
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
77806
Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
119657
If $ f _p(x)=1+x+x^2+ \cdots +x^{p-1} $ where $p$ is prime number, $e$ is any positive integer.