Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused machine learning model to measure the comparative regularity of prime number fields at specific regions of an Ulam spiral. Specifically, we demonstrate that in pure accuracy terms, models trained on blocks extracted from regions of the spiral in the vicinity of 500m outperform models trained on blocks extracted from the region representing integers lower than 25m. This implies existence of more easily learnable order in the former region than in the latter. Moreover, a detailed breakdown of precision and recall scores seem to imply that the model is favouring a different approach to classification in different regions of the spiral, focusing more on identifying prime patterns at lower numbers and more on eliminating composites at higher numbers. This aligns with number theory conjectures suggesting that at higher orders of magnitude we should see diminishing noise in prime number distributions, with averages (density, AP equidistribution) coming to dominate, while local randomness regularises after scaling by log x. Taken together, these findings point toward an interesting possibility: that machine learning can serve as a new experimental instrument for number theory. Notably, the method shows potential 1 for investigating the patterns in strong and weak primes for cryptographic purposes.

翻译：素数分布的研究揭示了一种双重特性：定义上具有确定性，却表现出类似随机过程的统计行为。本文表明，可以使用基于图像的机器学习模型来测量乌拉姆螺旋特定区域素数场的相对规律性。具体而言，我们证明在纯准确率指标上，使用从500m附近螺旋区域提取的数据块训练的模型，其性能优于使用从代表小于25m整数区域提取的数据块训练的模型。这意味着在前一区域中存在比后一区域更易于学习的秩序。此外，精确率和召回率的详细分析似乎表明，模型在螺旋不同区域倾向于采用不同的分类策略：在较小数字区域更注重识别素数模式，在较大数字区域更注重排除合数。这与数论猜想相一致，即在高数量级下素数分布的噪声应逐渐减弱，平均值（密度、等差数列均匀分布）将占主导地位，而局部随机性在按log x缩放后趋于规律化。综合来看，这些发现指向了一个有趣的可能性：机器学习可作为数论研究的新实验工具。值得注意的是，该方法在密码学应用中研究强素数与弱素数模式方面展现出潜力。