BiEntropy, TriEntropy and Primality

TL;DR

This paper introduces BiEntropy and TriEntropy as measures of order and disorder in number representations, revealing significant differences between primes and non-primes, and proposing implications for prime distribution and the Riemann Hypothesis.

Contribution

It presents novel entropy-based metrics for analyzing primes, demonstrating their discriminative power and deriving new bounds related to prime number theory.

Findings

BiEntropy distinguishes primes from non-primes with quadratic density.

TriEntropy correlates with BiEntropy, enabling prime classification.

Results suggest implications for the Riemann Hypothesis and twin primes conjecture.

Abstract

The order and disorder of binary representations of the natural numbers < 2^8 is measured using the BiEntropy function. Significant differences are detected between the primes and the non primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a monte carlo simulation for a sample of the natural numbers < 2^32 and in trinary for all natural numbers < 3^9 with similar but cubic results. We find a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical underpinnings of these results and show how they generalise to give a tight bound on the variance of Pi(x) - Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices.