Using Wilson's theorem, show that if $p$ is prime of the form $p=4n+1$ and if $y=(\frac {p-1}2)!$, then $y^2\equiv -1\mod p$.
19761
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.
88036
Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
107381
Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$
127816
Number theory proof mod p
126725
If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
22023
Infinitely number of primes in the form $4n+1$ proof
75318
How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
21629
Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
103360
Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
39915
Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
147101
Show that if $p>5$ is prime, then $240 \mid (p^4-1)$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
34692
Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$
20980
Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$