Let $p$ be a prime. Prove that $x^2\equiv 1\mbox{ mod }p$ if and only if $x\equiv 1\mbox{ mod }p$ or $x\equiv p-1\mbox{ mod }p$.
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
4129
Solvability of $a \equiv x^2 \mod b$
52530
There exists a prime congruent to $m$ mod $p$
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
23153
$\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$
86594
If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
111037
Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
126725
If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
94917
Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
64453
Prove if $(a,p)=1$, then $\{a,2a,3a,...,pa\}$ is a complete residue system modulo $p$.
102234
Show $a^p \equiv a \mod 2p$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.