Welcome to mathimatikoi.org; a site of university mathematics! Enjoy your stay here!

Does there exist expression?

Mathematical Competitions
Post Reply
User avatar
Riemann
Articles: 0
Posts: 164
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Does there exist expression?

#1

Post by Riemann » Sat Jul 22, 2017 9:33 am

For which positive integers $n$ does there exist expression

$$\mathbb{R}^2 = \bigcup_{m=1}^{\infty} A_m$$
disks.jpg
[/centre]

where each $A_m$ is a disk of radius $1$ such that each point $x \in \mathbb{R}^2$ belongs to either the boundary of some $A_m$ or to precisely $n$ interiors of the sets $A_1$ , $A_2$ , $\dots$ ?
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
Post Reply