BiEntropy, TriEntropy and Primality
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  • Release Date: 2021-11-15
  • prime number distribution
  • binary derivative
  • trinary derivative
  • Shannon entropy
Video Introduction

This video is adapted from 10.3390/e22030311

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. Researcher found a significant relationship between BiEntropy and TriEntropy such that he can discriminate between the primes and numbers divisible by six. Author discusses 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. He also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices.

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Croll, G. BiEntropy, TriEntropy and Primality. Encyclopedia. Available online: (accessed on 13 April 2024).
Croll G. BiEntropy, TriEntropy and Primality. Encyclopedia. Available at: Accessed April 13, 2024.
Croll, Grenville. "BiEntropy, TriEntropy and Primality" Encyclopedia, (accessed April 13, 2024).
Croll, G. (2021, November 15). BiEntropy, TriEntropy and Primality. In Encyclopedia.
Croll, Grenville. "BiEntropy, TriEntropy and Primality." Encyclopedia. Web. 15 November, 2021.