Let $n$ be an integer with $n \ge 2$. Show that $\phi(2^{n}-1)$ is divisible by $n$.

Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]

If $p$ is a prime and $n$ an integer such that $1<n \le p$, then \[\phi \left( \sum_{k=0}^{p-1}n^{k}\right) \equiv 0 \; \pmod{p}.\]

Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.

If $n$ is composite, prove that $\phi(n) \le n- \sqrt{n}$.

Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

Prove that for any $ \delta\in[0,1]$ and any $ \varepsilon>0$, there is an $ n\in\mathbb{N}$ such that $ \left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon$.

Show that the set of all numbers $\frac{\phi(n+1)}{\phi(n)}$ is dense in the set of all positive reals.

Show that if $n>49$, then there are positive integers $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1$. if $n>4$, then there are $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1$.

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

Determine all positive integers $n$ such that $n={d(n)}^2$.

Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.

Find all positive integers $n$ such that ${d(n)}^{3} =4n$.

Determine all positive integers for which $d(n)=\frac{n}{3}$ holds.

We say that an integer $m \ge 1$ is super-abundant if \[\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}\] for all $k \in \{1, 2,\cdots, m-1 \}$. Prove that there exists an infinite number of super-abundant numbers.

Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.

Prove that for any $\delta$ greater than 1 and any positive number $\epsilon$, there is an $n$ such that $\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon$.

Prove that $\sigma(n)\phi(n) < n^2$, but that there is a positive constant $c$ such that $\sigma(n)\phi(n) \ge c n^2$ holds for all positive integers $n$.

Show that $\sigma (n) -d(m)$ is even for all positive integers $m$ and $n$ where $m$ is the largest odd divisor of $n$.

Show that for any positive integer $n$, \[\frac{\sigma(n!)}{n!}\ge \sum_{k=1}^{n}\frac{1}{k}.\]

Let $n$ be an odd positive integer. Prove that $\sigma(n)^3 <n^4$.