MathOverflow

I asked a simillar question with the weaker restriction:
On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$
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I would like to know your comments in order to obtain the largest power of the prime numberr $p$ which divides$$F_p= \binom{p^{n+1}}{p^n}-\binom{p^{n}}{p^{n-1}}.$$I proved the largest power that divided $F_2$ is $3n$.For the case $p=2$, I could use Vandermonde...

For the function $f(x)=x^3-2x+2$, let $g(x)=x-\frac{f(x)}{f'(x)}$ and $M\subseteq \mathbb R$ be the set of points $c$ for which the sequence $(x_n)_{n\in\omega}$ with $x_0=c$ and $x_{n+1}=g(x_n)$ for $n\in\omega$ is convergent.
Problem. What is $\lim\limits_{R\to\infty}\frac{\lambda(M\cap[-R,R])}{2R}$?...

For the category of locally compact groups we know that there is a forgetful functor$$\mathbf{LocCptGrp}\longrightarrow\mathbf{DiscreteGrp} $$which sends $G$ to $G_d$, the discrete version of $G$. This functor "forgets" about the topology on $G$. Similarly...