For a positive integer $n$, prove that $\sum_{d|n}{\mu^2(d)/\phi(d)=n/\phi(n)}$
68385
Prove that $ \frac{n}{\phi(n)} = \sum\limits_{d \mid n} \frac{\mu^2(d)}{\phi(d)} $
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
90749
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
50927
If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
119118
Prove $n\mid \phi(2^n-1)$
100515
Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s φ-function.
84213
How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
47790
If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
58663
How to show that if $\phi(n)$ equals to n itself, then n must be 1?
16776
How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$
144819
Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
12976
Prove $\displaystyle\lim_{n\to\infty}\phi(n)=\infty$.
127047
How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
94558
Show that $\phi(mn) = \phi(m)\phi(n)\frac{d}{\phi(d)}$
44575
The sigma function (sum of divisors) multiplicative proof
98514
Prove by induction that the number of derangements of length $n$ is $D_n = (n-1)(D_{n-1}+D_{n-2}), n>2$