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Judging whether an integer can be divided by prime numbers such as 2 or 3 mayappear trivial to human beings, but can be less straightforward for computers.Here, we tested multiple deep learning architectures and feature engineeringapproaches on classifying integers based on their residues when divided bysmall prime numbers. We found that the ability of classification criticallydepends on the feature space. We also evaluated Automated Machine Learning(AutoML) platforms from Amazon, Google and Microsoft, and found that theyfailed on this task without appropriately engineered features. Furthermore, weintroduced a method that utilizes linear regression on Fourier series basisvectors, and demonstrated its effectiveness. Finally, we evaluated LargeLanguage Models (LLMs) such as GPT-4, GPT-J, LLaMA and Falcon, and demonstratedtheir failures. In conclusion, feature engineering remains an important task toimprove performance and increase interpretability of machine-learning models,even in the era of AutoML and LLMs.

Classification of integers based on residue classes via modern deep learning algorithms