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:

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

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

\begin{equation} N:=N+1 \end{equation}

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

\begin{equation} p_{n+1} = N \end{equation}

\begin{equation} \mathbb{P}^{n+1} := p_{n+1} \cup \mathbb{P}^n \end{equation}

\begin{equation} N := N+1 \end{equation}

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 \(U\):

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

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

\begin{equation} \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\} \end{equation}

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

\begin{equation} \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 \end{equation}

Moreover, it can be shown that:

\begin{equation} \log C = \text{span}(\log U) \end{equation}

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:

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

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:

\begin{equation} \Psi g(x) = g \circ \Psi(x) \end{equation}

which in a discrete setting implies:

\begin{equation} \Psi g(x_k) = g \circ \Psi(x_k) = g(x_{k+1}) \end{equation}

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

\begin{equation} \Psi \circ (g_1 + g_2)(x) = g_1 \circ \Psi(x) + g_2 \circ \Psi(x) \end{equation}

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:

\begin{equation} \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 \end{equation}

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

\begin{equation} \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\} \end{equation}

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

\begin{equation} \text{span}(\log \mathbb{P}^n) \subset \text{span}(\log \mathbb{P}) \end{equation}

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

\begin{equation} \text{span}(\log \mathbb{Q}_+) = \text{span}(\log \mathbb{P}) \end{equation}

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:

\begin{equation} \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\} \end{equation}

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:

\begin{equation} \forall n \in \mathbb{N}, f \circ n = n+1 \end{equation}

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:

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

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

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

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

\begin{equation} X_{n+1} = \Psi_n \circ X_n \end{equation}

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