## Motivation:

I recently managed to find a dynamical system that may be identified with an optimal prime-sieve. This deterministic dynamical system is interesting because its phase-space dimension is unbounded and it is highly sensitive to the initial conditions.

Moreover, it happens to have important consequences for the existence of computable prime formulas.

## An optimal prime-sieve:

The goal of an optimal prime sieve is to find all prime numbers less than or equal to $$N \in \mathbb{N}$$. A relatively simple solution for finding the first $$n$$ prime numbers $$\mathbb{P}^n = \{p_i\}_{i=1}^n$$ is given by the following program.

First, define the function $$f$$ that takes as inputs $$N \in \mathbb{N}$$ and $$\mathbb{P}^n = \{p_i\}_{i=1}^n$$ and computes:

$$f(N, \mathbb{P}^n) = \prod_{i=1}^n (N \mod p_i)$$

If $$f(N, \mathbb{P}^n)=0$$,

$$N:=N+1$$

Otherwise, if $$f(N, \mathbb{P}^n)$$ = 1, we execute the updates:

$$p_{n+1} = N$$

$$\mathbb{P}^{n+1} := p_{n+1} \cup \mathbb{P}^n$$

$$N := N+1$$

and we halt this process when the cardinality of $$\mathbb{P}^n$$ is as desired.

It is worth noting that $$f$$ is a prime formula and therefore a natural question is whether such a formula is computable with finite resources.

## The Buckingham-Pi theorem:

The Buckingham-Pi theorem states that if a dynamical system involves $$n$$ physical variables and $$n$$ also happens to be the number of fundamental dimensions involved then you need a system of scientific units whose cardinality is greater than or equal to $$n$$.

If you have a system of fundamental units $$U = \{u_i\}_{i=1}^n$$ and all the other physical units $$C$$ are derived from $$U$$:

$$\forall c \in C, \exists \alpha_i \in \mathbb{Z}, c = \prod_{i=1}^n = u_i^{\alpha_i}$$

Furthermore, if we define the n-dimensional vector space:

$$\text{span}(\log U) = \big\{ \sum_{i=1}^n \alpha_i \log u_i \lvert u_i \in U, \alpha_i \in \mathbb{Z} \big\}$$

the $$u_i$$ are fundamental if they are dimensionally independent in the sense that $$\forall u_i, u_{j \neq i} \in U$$, $$\log u_i \perp \log u_{j \neq i}$$:

$$\exists \alpha_i \in \mathbb{Z}, \sum_{i=1}^n \alpha_i \log u_i \iff \alpha_i = 1 \land \alpha_{j \neq i} = 0$$

Moreover, it can be shown that:

$$\log C = \text{span}(\log U)$$

as every element in $$C$$ has a unique factorisation in terms of $$U$$.

## The Koopman operator:

In principle, any physical system may be modelled as a discrete dynamical system:

$$\forall k \in \mathbb{Z}, x_{k+1} = \Psi \circ x_k$$

where, without loss of generality, $$x_k \in S \subset \mathbb{R}^n$$, $$k$$ is a discrete time index and $$T:S \rightarrow S$$ is a dynamic map. This representation is epistemologically sound as data collected from dynamical systems always comes in discrete-time samples.

Within this context, we may represent data as evaluations of functions of the state $$x_k$$, known as observables. In fact, if $$g: S \rightarrow \mathbb{R}$$ is an observable associated with the system then the collection of all such observables forms a vector space due to the Buckingham-Pi theorem.

Now, the Koopman operator $$\Psi$$ is a linear transform on this vector space given by:

$$\Psi g(x) = g \circ \Psi(x)$$

which in a discrete setting implies:

$$\Psi g(x_k) = g \circ \Psi(x_k) = g(x_{k+1})$$

where the linearity of the Koopman operator follows from the linearity of the composition operator:

$$\Psi \circ (g_1 + g_2)(x) = g_1 \circ \Psi(x) + g_2 \circ \Psi(x)$$

So we may think of the Koopman operator as lifting dynamics of the state space to the space of observables.

Furthermore, we may make the key observation that if $$\Psi$$ is of rank $$n$$ then $$\Psi$$ describes the evolution of a dynamical system with an n-dimensional phase-space.

## The prime numbers satisfy the Buckingham-Pi theorem:

It may be shown that $$\mathbb{P}$$ forms a fundamental system of units in the sense of Buckingham-Pi since the elements of $$\log \mathbb{P}$$ are dimensionally independent $$\forall p_j, p_{i \neq j} \in \mathbb{P}, \log p_{i \neq j} \perp \log p_j$$ in the sense that:

$$\exists \alpha_i \in \mathbb{Z}, \sum_{i=1}^\infty \alpha_i \log p_i = \log p_j \iff \alpha_j = 1 \land \alpha_{i \neq j} = 0$$

Furthermore, if we define the infinite-dimensional vector space:

$$\text{span}(\log \mathbb{P}) = \Big\{\sum_{i=1}^\infty \alpha_i \log p_i < \infty \lvert p_i \in \mathbb{P}, \alpha_i \in \mathbb{Z}\Big\}$$

and if we define the first $$n$$ primes, $$\mathbb{P}^n = \{p_i\}_{i=1}^n$$ then by definition:

$$\text{span}(\log \mathbb{P}^n) \subset \text{span}(\log \mathbb{P})$$

Moreover, due to the unique factorisation of the integers, it may be shown that:

$$\text{span}(\log \mathbb{Q}_+) = \text{span}(\log \mathbb{P})$$

## A dynamical system associated with prime sieving:

We are now ready to construct a nonlinear dynamical system that may be identified exactly with the prime sieve discussed earlier.

Assuming that $$\pi(n)$$ defines the number of primes for all $$n \geq 2$$, we may identify the first $$\pi(n)$$ prime numbers $$\mathbb{P}^{\pi(n)}$$ with the $$\pi(n)$$-dimensional vector space:

$$\text{span}(\log \mathbb{P}^{\pi(n)}) = \Big\{\sum_{i=1}^{\pi(n)} \alpha_i \log p_i \cdot \vec{e_i} \lvert p_i \in \mathbb{P}^{\pi(n)}, \alpha_i \in \mathbb{N}\Big\}$$

where $$\vec{e_i}$$ is the usual n-dimensional unit vector with components $$\delta_{ij}$$.

Now, given that the dynamics on the integers(our states) may be described by the linear model:

$$\forall n \in \mathbb{N}, f \circ n = n+1$$

we may construct a bijective mapping $$G$$ which lifts dynamics from the state-space $$\mathbb{N}$$ to the space of observables $$\text{span}(\log \mathbb{P}^{\pi(n)})$$ such that:

$$\forall n \in \mathbb{N} \exists! X_n \in \text{span}(\log \mathbb{P}^{\pi(n)}), X_n = G \circ n$$

where $$X = \{X_i\}_{i=1}^{\pi(n)}, X_i = \alpha_i \cdot \log p_i$$ assuming that:

$$\exists \alpha_i \in \mathbb{N}, \log n = \sum_{i=1}^{\pi(n)} \alpha_i \cdot \log p_i$$

Now, let’s suppose that there exists a linear map $$\Psi_n \in \mathbb{R}^{n \times n}$$ such that:

$$X_{n+1} = \Psi_n \circ X_n$$

Given the infinitude of primes, the phase-space associated with the evolution of $$X_n$$ is unbounded. It may also be demonstrated that this system is highly sensitive to the initial conditions.

## Discussion:

It is worth noting that since each dimension is orthogonal to the other dimensions, the number of bits required to describe this dynamical system is bounded by its dimensionality at instant $$n$$. Therefore, the Kolmogorov Complexity of this dynamical system is unbounded.

From this we may deduce that a prime formula would also have unbounded Kolmogorov Complexity.

## References:

1. Bernard Koopman. Hamiltonian systems and Transformations in Hilbert Space.
2. Steven L. Brunton. Notes on Koopman operator theory. 2019.
3. Fine & Rosenberger. Number Theory: An Introduction Via the Distribution of Primes. 2007.
4. Don Zagier. Newman’s short proof of the Prime Number Theorem. The American Mathematical Monthly, Vol. 104, No. 8 (Oct., 1997), pp. 705-708