Let $p$ be prime, $a$ an integer, $(a,p)=1$ and $n$ a positive integer $(n,p-1)=1$. Prove $x^n\equiv a$ mod $p$ has a unique solution.
121142
Take $G$ to be a group of order $600$. Prove that for any element a in $G$ there exist a $b$ in $G$ such that $a = b^7$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
52530
There exists a prime congruent to $m$ mod $p$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
102234
Show $a^p \equiv a \mod 2p$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
63356
Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
146312
Is the set of roots with respect to their values and multiplicies unique for each $g(x)$ in $a_n x^n + g(x)$?
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
4129
Solvability of $a \equiv x^2 \mod b$
119630
For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
21874
Solve $x^3+x \equiv 1 \pmod p$
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
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 $b^2\equiv a \mod p$.
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
94855
Prove if a (mod n) has a multiplicative inverse, then it's unique
126725
If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...