if $p \not= 2$ and $a^2 \equiv -1 \text{(mod p) }$ then $p \equiv 1 \text{(mod 4)}$
27287
$-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
26478
Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
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 $a$
140198
Let $a$ be a quadratic residue modulo $p$. Prove that the number $b\equiv a^\frac{p+1}{4} \mod p$ has the property that $b^2\equiv a \mod p$.
122741
$x^2$ is congruent to $-1 \bmod p$
23830
Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
33735
The square of an integer is congruent to 0 or 1 mod 4
61860
Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
76032
Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5