If $a\equiv 1\pmod {p^2},$ then $a\equiv 1\pmod p,$ for $\,a = r^{{\rm ord}(r)}$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
109532
Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
154933
Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
124995
$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
62239
For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
49166
$R^n \cong R^m$ iff $n=m$
15436
Find p and q s.t. $({p-1})^{q-1}+({q-1})^{p-1} \equiv 101 \pmod {pq}$
46746
How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
151985
If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
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 $b^2\equiv a \mod p$.
102234
Show $a^p \equiv a \mod 2p$