Certain tasks such as determining whether a given integer can be divided by 2, 3, or other prime numbers may be trivial for human beings, but can be less straightforward for computers in the absence of pre-specified algorithms. In this paper, we tested multiple deep learning architectures and feature engineering approaches, and evaluated the scenario of determining divisibility of large finite integers (up to $2^{32}$) by small prime numbers. It turns out that, regardless of the network frameworks or the complexity of the network structures (CNN, RNN, Transformer, etc.), the ability to predict the prime number divisibility critically depends on the feature space fed into the deep learning models. We also evaluated commercially available Automated Machine Learning (AutoML) pipelines from Amazon, Google and Microsoft, and demonstrated that they failed to address this issue unless appropriately engineered features were provided. We further proposed a closed form solution to the problem using the ordinary linear regression on Fourier series basis vectors, and showed its success. Finally, we evaluated prompt-based learning using ChatGPT and demonstrated its success on small primes and apparent failures on larger primes. We conclude that feature engineering remains an important task to improve the performance, increase the interpretability, and reduce the complexity of machine learning/deep learning models, even in the era of AutoML and large-language models (LLMs).

本研究测试多个深度学习架构和特征工程方法，评估了使用有限整数（< 2^32）除以小质数的情况，提出一种基于傅里叶级数基向量的闭合解决方案，并且证明了自动机器学习（AutoML）流程需要适当的特征工程来处理该问题。同时，我们的结果表明，即使在AutoML和LLM时代，特征工程仍然是提高性能、增加可解释性和减少机器学习/深度学习模型复杂性的重要任务。

深度学习在质数可除性方面的应用