USING IFS TO REVEAL BIASES IN THE DISTRIBUTION OF PRIME NUMBERS

Fractals Pub Date : 2024-07-16 DOI:10.1142/s0218348x24501032
Harlan J. Brothers
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Abstract

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias.
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用ifs揭示质数分布的偏差
长期以来,人们一直认为素数的伪随机分布不存在偏差。具体来说,虽然素数定理给出了找到素数概率的渐近度量,而关于算术级数的狄利克特定理也告诉了我们素数在不同残差类别中的分布,但我们没有理由相信连续素数之间会 "了解 "彼此--例如,它们可能倾向于避免以相同的数字结尾。在这里,我们展示了迭代函数系统方法(IFS)是一种令人惊讶的有用工具,它可以揭示这种不直观的结果,并更广泛地研究数论中的结构。我们在 2013 年的一项研究中得出的实验结果包括分形模式,它揭示了具有特定同位性质的各类素数之间的 "排斥 "现象。我们的计算和解释中显示的一些现象与莱姆克-奥利弗和桑达拉拉詹最近关于连续素数之间偏差的研究有关。在这里,我们通过展示 IFS 如何从动态的角度精确地指出这些偏差的行为方式,来探索和扩展这些成果。我们还证明,令人惊讶的是,复合数也会表现出明显类似的偏差。
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