Uncovering multiscale order in the prime numbers via scattering
Uncovering multiscale order in the prime numbers via scattering
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DOI：
10.1088/1742-5468/aad6be
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
S. Torquato;G. Zhang;M. D. Courcy-Ireland
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文献类型：
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作者：
S. Torquato;G. Zhang;M. D. Courcy-Ireland
几千年来，质数一直是令人着迷的地方，并继续给我们带来惊喜。在最近物理和材料科学领域引起关注的超均匀概念的启发下，我们证明了某些大区间中的素数在长度尺度上具有意想不到的阶数，并且代表了一类新的具有纯点衍射图案的多粒子系统的第一个例子，我们称之为有效的极限周期。特别是，这种体制中的质数是超级一致的。这是使用结构因子解析地显示出来的，它与来自多粒子系统的散射强度成正比。值得注意的是，素数的结构因子以致密的布拉格峰为特征，类似于准晶，但位于某些有理波数处，类似于极限周期点图案。然而，素数显示了占用和未占用站点的不稳定模式，这与标准极限周期系统的可预测模式非常不同。我们还确定了有序和无序素数制度之间的转变，这取决于所研究的区间。我们的分析导致了一种算法，使人们能够高精度地预测素数。有效极限周期值得在物理学中进一步研究，而不考虑它与素数的联系。
The prime numbers have been a source of fascination for millennia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor , proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor of the primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. However, the primes show an erratic pattern of occupied and unoccupied sites, very different from the predictable patterns of standard limit-periodic systems. We also identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.