Prove that if the $\mathrm{ord}(p)a =3$, then $\mathrm{ord}(p)(a+1) = 6$, $p$ prime
61860
Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.
33714
$p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
85287
Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$
103496
Show that $3p^2=q^2$ implies $3|p$ and $3|q$
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
131488
Show that $p^3+4$ is prime
74187
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
128575
Elementary proof that $3$ is a primitive root of a Fermat prime?
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
100892
Proving divisibility of $a^3 - a$ by $6$
125188
how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
100984
Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
138941
If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
151985
If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
37093
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$