Prove that if there exists an integer a such that $a^2 \equiv −1 \text{ (mod p) then } p ≡ 1 \text{ (mod 4)}$
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
151985
If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
27063
Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
157711
Number Theory: Prove that $m | \phi(a^m - 1)$
36043
If $ab \equiv r \pmod{p}$, and $x^2 \equiv a \pmod{p}$ then $y^2 \equiv b \pmod{p}$ for which condition of $r$?
6160
How to prove a number $a$ is quadratic residue modulo $n$?
18108
Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
21629
Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
104484
Prove that $r$ is a primitive root modulo $p$ if and only if $r^{(p−1)/q}\not\equiv 1\pmod{p}$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
75318
How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
146053
Proving identities (mod $pq$) using Fermat's little theorem?
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
124995
$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$