if $L \geq 2$ has the property that whenever $n\equiv m\mod L$, we have $n^n \equiv m^m\mod p$, then $p(p - 1) \leq L.$
100696
Let $p$ be prime. Consider the sequence: $1, 2^2, 3^3, 4^4, 5^5, 6^6 \cdots \pmod p$. Prove the sequence is periodic with smallest period $p(p-1).$
122612
An integer is a $n$'th power if that holds true for all moduli.
108237
Show $a^p \equiv b^p \mod p^2$
106401
Prove: $p-mq | f(m)$ where $m$ is any integer
15314
Number of solutions to $x^n \equiv 1 \mod p$
64239
Upper Bound for $p(1-p) $ where $0 \le p \le 1$
110752
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
19804
How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
154933
Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
144693
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
19071
solution set for congruence $x^2 \equiv 1 \mod m$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
114867
Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$