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.

中文翻译：

---

双熵、三熵和素性

使用 BiEntropy 函数测量自然数 < 2^8 的二进制表示的顺序和无序。在素数和非素数之间检测到显着差异。BiEntropic 素数密度显示为二次的，具有非常小的高斯分布误差。使用蒙特卡罗模拟对自然数 < 2 ^ 32 的样本进行二进制重复该工作，对所有 < 3 ^ 9 的自然数进行三进制重复，结果相似但三次。我们发现 BiEntropy 和 TriEntropy 之间存在显着关系，因此我们可以区分质数和可被 6 整除的数。我们讨论了这些结果的理论基础，并展示了它们如何概括以给出所有 x 的 Pi(x) - Li(x) 方差的严格界限。这个界限比 Von Koch 在 1901 年作为证明黎曼假设的等价给出的界限要严格得多。由于对二元导数的简单归纳，素数是高斯的，这意味着孪生素数猜想是正确的。我们还在附录中提供了费马素数和梅森素数的绝对收敛渐近线。