Fractals and discrete dynamics associated to prime numbers

Abstract

Abstract On this work, it is shown that fractal theory can be applied in number theory to analyze the distribution of prime numbers. Based on the similarities between the distribution of prime numbers’ powers for each natural number and generalized Cantor’s sets, some symmetric properties are analyzed and used to propose a set of discrete dynamics that allow visualizing the recursive and symmetric properties of prime numbers, which leads to the proposition of an approximation of a fractal version of the distribution function of prime numbers. Then, with this set of discrete dynamics, it is proposed a prime numbers’ sieve algorithm that seems to describe the growth of a fractal and its fractal dimension is computed, suggesting that the fractal and chaotic behavior on the distribution of prime numbers emerges from: 1) the increase of the memory or dimension of the discrete dynamics, that introduces new frequencies of oscillation each stage and 2) the symmetry break due to the exponential growth of the length of the symmetries with respect to the domain of applicability of each discrete dynamics. Therefore, although chaotic, the distribution of prime numbers has a deterministic chaotic behavior, with symmetries and harmonics, that follows the classical path to chaos, i.e. from periodic to quasi periodic to chaos.