[Paper Review] Fractal in the statistics of Goldbach partition

Liang Wang, Huang Yan+1 more|ArXiv.org|Jan 12, 2006

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper investigates the statistical distribution of Goldbach partitions, where even numbers are expressed as sums of two primes, revealing fractal-like periodic oscillations in the count r(n) of such representations. Using symbolic dynamics and the Hardy-Littlewood conjecture, it demonstrates that r(n) exhibits self-similar, multi-level periodic structures across different scales, indicating a deep fractal organization in the additive structure of prime numbers.

ABSTRACT

Some interesting chaos phenomena have been found in the difference of prime numbers. Here we discuss a theme about the sum of two prime numbers, Goldbach conjecture. This conjecture states that any even number could be expressed as the sum of two prime numbers. Goldbach partition r(n) is the number of representations of an even number n as the sum of two primes. This paper analyzes the statistics of series r(n) (n=4,6,8,...). The familiar 3 period oscillations in histogram of difference of consecutive primes appear in r(n).We also find r(n) series could be divided into different levels period oscillation series. The series in the same or different levels are all very similar, which presents the obvious fractal phenomenon. Moreover, symmetry between the statistics figure of sum and difference of two prime numbers are also described. We find the estimate of Hardy-Littlewood could precisely depict these phenomena. A rough analyzing for periodic behavior of r(n) is given by symbolic dynamics theory at last.

Motivation & Objective

• To analyze the statistical distribution of Goldbach partitions r(n), defined as the number of ways even integers n can be written as the sum of two primes.
• To investigate the presence of periodic and self-similar structures in r(n) across different scales.
• To explore the symmetry between the statistics of sums and differences of prime pairs.
• To determine whether the Hardy-Littlewood conjecture accurately models the observed fractal-like oscillations in r(n).

Proposed method

• The authors compute r(n) for even integers n = 4, 6, 8, ..., and analyze its histogram to detect periodic patterns.
• They identify 3-period oscillations in r(n), similar to those observed in the differences of consecutive primes.
• The data is segmented into distinct 'levels' of periodic oscillations, revealing hierarchical, self-similar structures suggestive of fractal behavior.
• Symbolic dynamics theory is applied to provide a rough theoretical explanation for the periodic behavior in r(n).
• The Hardy-Littlewood conjecture is used to estimate r(n), and its accuracy in capturing the observed oscillations is evaluated.
• Visual and statistical analysis of the data is performed using 11 figures to illustrate symmetry and scaling patterns.

Experimental results

Research questions

• RQ1 Does the number of Goldbach partitions r(n) exhibit periodic oscillations across different scales of n?
• RQ2 Are the observed oscillations in r(n) self-similar, indicating fractal-like behavior?
• RQ3 How does the symmetry between sums and differences of prime pairs manifest in the statistical distribution of r(n)?
• RQ4 To what extent can the Hardy-Littlewood conjecture predict the observed periodic and fractal structures in r(n)?
• RQ5 Can symbolic dynamics provide a theoretical framework for understanding the periodicity in r(n)?

Key findings

• The r(n) sequence displays clear 3-period oscillations, mirroring similar patterns in the differences of consecutive primes.
• The r(n) series can be decomposed into multiple levels of periodic oscillations, each exhibiting self-similarity across scales, indicating fractal structure.
• Different levels of oscillation in r(n) are highly similar in shape, reinforcing the presence of fractal-like organization.
• A strong symmetry is observed between the statistical distributions of sums and differences of prime pairs.
• The Hardy-Littlewood conjecture provides a precise estimate that accurately captures the observed periodic and fractal patterns in r(n).
• Symbolic dynamics offers a theoretical basis for understanding the emergence of periodic behavior in r(n), suggesting a dynamical system origin for the observed structures.