*This question came to my mind thanks to this question which I found really interesting (and beautiful! Like the mathematician Philippe Caldero said in his book Histoires Hédonistes de Groupes et de Géométries (roughly translated) "Let us stop for a moment to contemplate the beauty of mathematics, that is after all the point of figures".).*

*It is also related to this other question.*

The idea is to perform a walk following the following rules:

Initialisation: You start on the point $(0,0)$ which correspond to the integer $n=0$, and you will walk from one point of $\mathbb Z^2$ to another. You start by walking on the right.

Each horizontal step you take increases the integer $n$ by $1$.

When $n$ is equal to a prime number, you take one step up, and you change the direction you were going to (if you were walking from left to right you will walk from right to left, and reciprocally).

To illustrate the rules, a drawing will perhaps be more explicit:

$$\begin{matrix} & 7 & 8 & 9 & 10\\ & 7 & 6 & 5 \\ & 3 & 4 & 5 \\ & 3 & 2 & \\ 0&1&2& & \end{matrix}$$

Or with blue lines:

Well now nothing stops us from going a little further, which we will do until $n=100$, and then until $n=1\,000$.

It seems that the walk is almost always on the right side of the $y$-axis. Though the walk is crossing the axis a few times.

Let us walk until $n=10\,000$.

Then we realise that we have completely changing the side of the axis we were walking on.

Which rises some questions:

Will we cross the $y$-axis infinitely many times?

Will we be walking as much on each side of the plane? In the sense that if we denote by $L_n$ the set of integers less or equals to $n$ on the left side of the plane and $R_n$ the set of integers less or equals to $n$ on the left side of the plane:

$$\lim_{n\to\infty} \frac {L_n}n=\lim_{n\to\infty} \frac {R_n}n=\frac 12.$$

- Will we walk out of any fixed vertical band centred on the $y$-axis?

Though any other result, or **drawing** (I did not succeed in drawing it for $n=10^5$), references about this walk would be of great interest.