If we plot $f(n) = \dfrac{n+1}{p\_{n+1}} - \dfrac{n}{p\_n}, n \in \mathbb{N} $, we get an interesting pattern.

[![enter image description here](https://i.stack.imgur.com/6EFNh.png)](https://i.stack.imgur.com/6EFNh.png)

Questions:

* Why it looks like there's different lines on plot?
* Why they have such shape?
* How those lines can be approximated?
* Why there's like peak at $n \approx 500$?
* Why points in lines become sparcer as they are closer to the bottom?