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

Area of Science:

• Number Theory
• Computer Science
• Artificial Intelligence

Background:

• Classifying integers by prime number residues is a fundamental computational task.
• Deep learning and automated machine learning (AutoML) approaches are increasingly used for complex classification problems.
• The performance of machine learning models often hinges on the quality of input features.

Purpose of the Study:

• To investigate the effectiveness of various deep learning architectures and feature engineering techniques for classifying integers based on their residues modulo small primes.
• To evaluate the performance of leading AutoML platforms on this specific number-theoretic task.
• To assess the capabilities of large language models (LLMs) in handling such classification problems and to introduce a novel, effective method.

Main Methods:

• Testing diverse deep learning architectures and feature engineering strategies.
• Evaluating AutoML platforms from Amazon, Google, and Microsoft.
• Developing and applying a novel method using linear regression on Fourier series basis vectors.
• Assessing the performance of large language models (LLMs) including GPT-4, GPT-J, LLaMA, and Falcon.

Main Results:

• Classification performance is highly dependent on the chosen feature space.
• AutoML platforms failed to perform the task without expert feature engineering.
• The proposed linear regression method on Fourier series basis vectors proved effective.
• Large language models (LLMs) demonstrated significant limitations in this classification task.

Conclusions:

• Feature engineering remains a critical component for enhancing machine learning model performance and interpretability.
• Despite advancements in AutoML and LLMs, tailored feature engineering is essential for specific, complex tasks.
• The study highlights the enduring importance of fundamental machine learning principles alongside cutting-edge technologies.