BiEntropy, TriEntropy and Primality

Abstract

The order and disorder of binary representations of the natural numbers < 28 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 of a sample of natural numbers < 232 and in trinary for all natural numbers < 39 with similar but cubic results. We found a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical basis 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.