Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

Interpretable Architecture Neural Networks for Function Visualization

In many scientific research fields, understanding and visualizing a black-box function in terms of the effects of all the input variables is of great importance. Existing visualization tools do not allow one to visualize the effects of all the input variables simultaneously. Although one can select one or two of the input variables to visualize via a 2D or 3D plot while holding other variables fixed, this presents an oversimplified and incomplete picture of the model. To overcome this shortcoming, we present a new visualization approach using an interpretable architecture neural network (IANN) to visualize the effects of all the input variables directly and simultaneously. We propose two interpretable structures, each of which can be conveniently represented by a specific IANN, and we discuss a number of possible extensions. We also provide a Python package to implement our proposed method. The supplemental materials are available online.