Using Dynamical Systems to Construct Infinitely Many Primes

TL;DR

This paper explores how ideas from arithmetic dynamics can be used to construct infinitely many primes, offering new methods and formulas inspired by Euclid's classical proof.

Contribution

It introduces a novel approach combining dynamical systems with prime construction, including a converging formula for primes.

Findings

New prime construction methods using dynamical systems

A fast converging formula for primes involving limits and products

Extension of Euclid's classical proof with modern mathematical tools

Abstract

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. After acceptance Soundararajan noted the beautiful and fast converging formula: $$ \tau = a^{1/(d-1)} x_0 \cdot \lim_{n\to \infty} \prod_{m=1}^n \left(\frac{x_m}{ax_{m-1}^d} \right)^{1/d^m} $$