The answer given are already very good. I just wanted to point, that there are also an infinte family of metrics who embed in euclidean space.

First notice that for finite sets $X,Y$ contained in a larger finite set $Z$, the symmetric difference metric $d(X,Y) = \sqrt{|X|+|Y|-2|X \cap Y|}$ can be embedded in euclidean space by listing the elements of $Z$ in an ordered way and the vector $\phi(X)$ is a binary vector with the $i$-th entry $=1$ if $z_i \in X$ and $0$ otherwise. Then $|\phi(X)| = |X|$ and $|\phi(X)-\phi(Y)|^2 = d(X,Y)$, which shows the embedding.

By considering the sets $X_a = \{ a/k | 1 \le k \le a \}$ and noticing that $|X_a \cap X_b| = \gcd(a,b)$ we get the metric on natural numbers:

$d(a,b) = \sqrt{|X_a|+|X_b|-2|X_a\cap X_b|} = \sqrt{a+b-2 \gcd(a,b)}$ which can be embedded as was shown in Euclidean space.

On the other hand if we consider sets $X_a$ such that $|X_a| = \sigma_k(a)$ where $k \ge 0$ and $\sigma_k(a) = \sum_{d|a}d^k$, which are not difficult to construct, and such that $|X_a \cap X_b| = \sigma(\gcd(a,b))$, we get an infinte sequences of metrics, which can be embedded in Euclidean space:

$$d_{\sigma,k}(a,b) = \sqrt{\sigma_k(a)+\sigma_k(b)-2\sigma_k(\gcd(a,b))}$$. For $k=0$ and $\tau(a) = \sigma_0(a)$ we observe,that primes $p$ have norm equal to one:

$$|p|:= d_{\sigma,0}(1,p) = \sqrt{1+2-2\cdot 1}=1$$

Hence under this metric all primes are on the $1$-sphere.

Especially for $k=1$ and $p,q,r$ three pairwise distinct primes, we get invoking the law of cosines and $d_1(p,q)^2 = p+q$ the following nice formula:

$$\pi = \operatorname{acos}(\frac{r}{\sqrt{(p+r)(q+r)}})+\operatorname{acos}(\frac{q}{\sqrt{(p+q)(q+r)}})+\operatorname{acos}(\frac{p}{\sqrt{(p+r)(p+q)}})$$

The answer given are already very good. I just wanted to point, that there are also an infinte family of metrics who embed in euclidean space.

First notice that for finite sets $X,Y$ contained in a larger finite set $Z$, the symmetric difference metric $d(X,Y) = \sqrt{|X|+|Y|-2|X \cap Y|}$ can be embedded in euclidean space by listing the elements of $Z$ in an ordered way and the vector $\phi(X)$ is a binary vector with the $i$-th entry $=1$ if $z_i \in X$ and $0$ otherwise. Then $|\phi(X)| = |X|$ and $|\phi(X)-\phi(Y)|^2 = d(X,Y)$, which shows the embedding.

By considering the sets $X_a = \{ a/k | 1 \le k \le a \}$ and noticing that $|X_a \cap X_b| = \gcd(a,b)$ we get the metric on natural numbers:

$d(a,b) = \sqrt{|X_a|+|X_b|-2|X_a\cap X_b|} = \sqrt{a+b-2 \gcd(a,b)}$ which can be embedded as was shown in Euclidean space.

On the other hand if we consider sets $X_a$ such that $|X_a| = \sigma_k(a)$ where $k \ge 0$ and $\sigma_k(a) = \sum_{d|a}d^k$, which are not difficult to construct, and such that $|X_a \cap X_b| = \sigma(\gcd(a,b))$, we get an infinte sequences of metrics, which can be embedded in Euclidean space:

$$d_{\sigma,k}(a,b) = \sqrt{\sigma_k(a)+\sigma_k(b)-2\sigma_k(\gcd(a,b))}$$. For $k=0$ and $\tau(a) = \sigma_0(a)$ we observe,that primes $p$ have norm equal to one:

$$|p|:= d_{\sigma,0}(1,p) = \sqrt{1+2-2\cdot 1}=1$$

Hence under this metric all primes are on the unit sphere.

Especially for $k=1$ and $p,q,r$ three pairwise distinct primes, we get invoking the law of cosines and $d_1(p,q)^2 = p+q$ the following nice formula:

$$\pi = \operatorname{acos}(\frac{r}{\sqrt{(p+r)(q+r)}})+\operatorname{acos}(\frac{q}{\sqrt{(p+q)(q+r)}})+\operatorname{acos}(\frac{p}{\sqrt{(p+r)(p+q)}})$$

The answer given are already very good. I just wanted to point, that there are also an infinte family of metrics who embed in euclidean space.

First notice that for finite sets $X,Y$ contained in a larger finite set $Z$, the symmetric difference metric $d(X,Y) = \sqrt{|X|+|Y|-2|X \cap Y|}$ can be embedded in euclidean space by listing the elements of $Z$ in an ordered way and the vector $\phi(X)$ is a binary vector with the $i$-th entry $=1$ if $z_i \in X$ and $0$ otherwise. Then $|\phi(X)| = |X|$ and $|\phi(X)-\phi(Y)|^2 = d(X,Y)$, which shows the embedding.

By considering the sets $X_a = \{ a/k | 1 \le k \le a \}$ and noticing that $|X_a \cap X_b| = \gcd(a,b)$ we get the metric on natural numbers:

$d(a,b) = \sqrt{|X_a|+|X_b|-2|X_a\cap X_b|} = \sqrt{a+b-2 \gcd(a,b)}$ which can be embedded as was shown in Euclidean space.

On the other hand if we consider sets $X_a$ such that $|X_a| = \sigma_k(a)$ where $k \ge 0$ and $\sigma_k(a) = \sum_{d|a}d^k$, which are not difficult to construct, and such that $|X_a \cap X_b| = \sigma(\gcd(a,b))$, we get an infinte sequences of metrics, which can be embedded in Euclidean space:

$$d_{\sigma,k}(a,b) = \sqrt{\sigma_k(a)+\sigma_k(b)-2\sigma_k(\gcd(a,b))}$$. For $k=0$ and $\tau(a) = \sigma_0(a)$ we observe,that primes $p$ have norm equal to one:

$$|p|:= d_{\sigma,0}(1,p) = \sqrt{1+2-2\cdot 1}=1$$

Hence under this metric all primes are on the $1$-sphere.

The answer given are already very good. I just wanted to point, that there are also an infinte family of metrics who embed in euclidean space.

First notice that for finite sets $X,Y$ contained in a larger finite set $Z$, the symmetric difference metric $d(X,Y) = \sqrt{|X|+|Y|-2|X \cap Y|}$ can be embedded in euclidean space by listing the elements of $Z$ in an ordered way and the vector $\phi(X)$ is a binary vector with the $i$-th entry $=1$ if $z_i \in X$ and $0$ otherwise. Then $|\phi(X)| = |X|$ and $|\phi(X)-\phi(Y)|^2 = d(X,Y)$, which shows the embedding.

By considering the sets $X_a = \{ a/k | 1 \le k \le a \}$ and noticing that $|X_a \cap X_b| = \gcd(a,b)$ we get the metric on natural numbers:

$d(a,b) = \sqrt{|X_a|+|X_b|-2|X_a\cap X_b|} = \sqrt{a+b-2 \gcd(a,b)}$ which can be embedded as was shown in Euclidean space.

On the other hand if we consider sets $X_a$ such that $|X_a| = \sigma_k(a)$ where $k \ge 0$ and $\sigma_k(a) = \sum_{d|a}d^k$, which are not difficult to construct, and such that $|X_a \cap X_b| = \sigma(\gcd(a,b))$, we get an infinte sequences of metrics, which can be embedded in Euclidean space:

$$d_{\sigma,k}(a,b) = \sqrt{\sigma_k(a)+\sigma_k(b)-2\sigma_k(\gcd(a,b))}$$. For $k=0$ and $\tau(a) = \sigma_0(a)$ we observe,that primes $p$ have norm equal to one:

$$|p|:= d_{\sigma,0}(1,p) = \sqrt{1+2-2\cdot 1}=1$$

Hence under this metric all primes are on the $1$-sphere.

Especially for $k=1$ and $p,q,r$ three pairwise distinct primes, we get invoking the law of cosines and $d_1(p,q)^2 = p+q$ the following nice formula:

$$\pi = \operatorname{acos}(\frac{r}{\sqrt{(p+r)(q+r)}})+\operatorname{acos}(\frac{q}{\sqrt{(p+q)(q+r)}})+\operatorname{acos}(\frac{p}{\sqrt{(p+r)(p+q)}})$$