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BiEntropy, TriEntropy and Primality.

Grenville J Croll1

  • 1Alternative Natural Philosophy Association, Bury St Edmunds IP30 9QX, UK.

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Summary
This summary is machine-generated.

The BiEntropy function reveals distinct patterns in prime numbers, showing a quadratic prime density. This research implies the twin primes conjecture is true and offers new bounds for prime number distribution.

Keywords:
Shannon entropybinary derivativeprime number distributiontrinary derivative

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Area of Science:

  • Number Theory
  • Computational Mathematics
  • Information Theory

Background:

  • Understanding the distribution of prime numbers is a fundamental problem in mathematics.
  • Existing methods for analyzing prime distribution have limitations, particularly in providing tight bounds.

Purpose of the Study:

  • To investigate the order and disorder of natural numbers using the BiEntropy function.
  • To explore the relationship between number representations and prime number properties.
  • To derive new bounds for prime number distribution and test conjectures like the twin primes conjecture.

Main Methods:

  • Application of the BiEntropy function to binary representations of natural numbers (< 2^8).
  • Monte Carlo simulations for larger number sets (binary < 2^32, trinary < 3^9).
  • Analysis of the relationship between BiEntropy and TriEntropy for number classification.
  • Generalization of findings to derive bounds for Pi(x)-Li(x).

Main Results:

  • Significant differences detected between primes and non-primes using BiEntropy.
  • BiEntropic prime density shown to be quadratic with minimal Gaussian error.
  • Similar quadratic and cubic results observed in binary and trinary systems.
  • A tight bound on the variance of Pi(x)-Li(x) derived, surpassing previous bounds.
  • The study implies the twin primes conjecture is true due to primes being Gaussian.

Conclusions:

  • The BiEntropy function provides a novel method for analyzing prime number distribution.
  • The findings offer significant theoretical implications for number theory, including the twin primes conjecture.
  • The derived bounds for prime distribution are substantially tighter than previously established ones.