Multiple proofs of $\sum_{d|n}{\phi(d)}=n$
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
26658
A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
32780
Combinatorial Proof Question: Vandermonde convolution identity
115258
Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
19599
$ d(x_n,y_n) \to d(x,y)$ if $x_n\to x$ and $y_n \to y$
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
49938
Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
48251
Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.
29828
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
76347
Shorter proof for some equvalences
98514
Prove by induction that the number of derangements of length $n$ is $D_n = (n-1)(D_{n-1}+D_{n-2}), n>2$
137982
Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$
125188
how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
144819
Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
37012
The number of summands $\phi(n)$
27901
Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$
17241
How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$