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

Judging whether an integer can be divided by prime numbers such as 2 or 3 may appear trivial to human beings, but it can be less straightforward for computers. Here, we tested multiple deep learning architectures and feature engineering approaches to classifying integers based on their residues when divided by small prime numbers. We found that the ability of classification critically depends on the feature space. We also evaluated automated machine learning (AutoML) platforms from Amazon, Google, and Microsoft and found that, without appropriately engineered features, they failed on this task. Furthermore, we introduced a method that utilizes linear regression on Fourier series basis vectors and demonstrated its effectiveness. Finally, we evaluated large language models (LLMs) such as GPT-4, GPT-J, LLaMA, and Falcon, and we demonstrated their failures. In conclusion, feature engineering remains an important task to improve performance and increase interpretability of machine learning models, even in the era of AutoML and LLMs.