-1 as a Quadratic Residue mod $p$ $\Rightarrow$ $p \equiv_4 1$
101349
$-1$ isn't a quadratic residue $\pmod{p}, p=4k+3$.
78379
Proof on cubic residues
126725
If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
130162
A complete residue system from an arithmetic progression
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
60780
Help needed in understanding proof: Every odd prime $p$ has exactly $(p-1)/2$ quadratic residues and $(p-1)/2$ quadratic nonresidues.
122741
$x^2$ is congruent to $-1 \bmod p$
52530
There exists a prime congruent to $m$ mod $p$
102234
Show $a^p \equiv a \mod 2p$
146851
Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$
13317
show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ such that $\text{gcd}(a, 32760) = 1$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
6160
How to prove a number $a$ is quadratic residue modulo $n$?
25910
Product of Quadratic Residues
153862
A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
127291
Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$