Prove $a^m\equiv a^{m-\phi(m)}\pmod m$ for all positive integers
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
108237
Show $a^p \equiv b^p \mod p^2$
49938
Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
115258
Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
119118
Prove $n\mid \phi(2^n-1)$
154933
Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
145397
How to prove that for every $i$, $a_i = i$
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
29462
Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
52951
Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
123891
There are finitely many maps on nonnegative integers satisfying $\phi(ab)=\phi(a)+\phi(b)$
6410
Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
27063
Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
139383
Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.