If a is a $\mathbb Z / p\mathbb Z$ generator, if $k$ is not a multiple of $p-1$, $a ^ k \not\equiv 1\pmod p$
4972
Mod of numbers with large exponents [modular order reduction]
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
88036
Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
51465
New, elegant proofs for $\varphi(p^{k})=p^{k}-p^{k-1}$
92211
When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?
102112
Idempotents in $\mathbb Z_n$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
97196
$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
118629
$p,q$ odd primes, $q\mid a^p-1\Rightarrow q\mid a-1$ or $\,2p\mid q-1$
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
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.
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
128566
Proving $2$ generates $(\mathbb{Z}/19\mathbb{Z})^*$ by only looking at a few powers
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$