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Entropy, Periodicity and the Probability of Primality.

Grenville J Croll1

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

Entropy (Basel, Switzerland)
|December 24, 2025
PubMed
Summary
This summary is machine-generated.

This study reveals that periodic numbers are composite and introduces a new entropic measure for primality. This finding deepens our understanding of prime number distribution and its connection to entropy.

Keywords:
BiEntropyRiemann Hypothesisbinary derivativeperiodicityprime number distributionprobability of primalityrandomness and complexity

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Area of Science:

  • Number Theory
  • Computational Mathematics
  • Information Theory

Background:

  • The distribution of prime numbers exhibits characteristics of both order and randomness.
  • Understanding the underlying structure of prime numbers is crucial for various mathematical and computational fields.

Purpose of the Study:

  • To investigate the relationship between entropy, periodicity, and primality using the binary derivative framework.
  • To introduce a novel entropic measure for determining primality.
  • To explore the implications of these findings for number theory and cryptography.

Main Methods:

  • Utilizing the computational framework of the binary derivative.
  • Proving theorems on the behavior of primes and composites related to binary periodicity.
  • Introducing a scale-invariant entropic measure of primality, p(s').

Main Results:

  • Periodic numbers are proven to be composite, with a single trivial exception.
  • A novel entropic measure, p(s'), is introduced, providing an exact probability of primality based on binary structure.
  • p(s') is shown to be quadratic, statistically sound, and correlated with binary disorder.
  • Empirical analyses indicate that prime density variance is small and binormal, constrained by the central limit theorem.

Conclusions:

  • A deep connection exists between entropy and the randomness of prime numbers.
  • The findings offer new insights into the entropic structure of number theory.
  • Potential implications for the Riemann Hypothesis, special prime classes, and cryptographic applications.