prove that $\phi(xy) =\phi(x)\phi(y)$ for any $x$ and $y$ with $(x, y) = 1$.
58188
What's the proof that the Euler totient function is multiplicative?
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
54266
Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$
84930
How to justify $\phi(n) \ge \sqrt{n}$
141166
How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$ is injective, then $\phi$ is also injective?
81671
Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
2455
Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
9753
How to solve the equation $\phi(n) = k$?
157711
Number Theory: Prove that $m | \phi(a^m - 1)$
102175
$Z=XY$, $E_Z[Z]$, $E_{X,Y}[XY]$
26658
A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
51483
Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$.
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
66457
Series expansion of $xy'=y$
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
36374
Exercise 1.6.3 from Alon & Spencer's *The Probabilistic Method*: prove that $Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1]$ for i.i.d. real RVs $X$ and $Y$