For decades, ongoing efforts have shown an increase in the use of artificial intelligence (AI) methods to accelerate scientific discoveries. However, accelerating discoveries in mathematics remains a persistent challenge for AI. Specifically, AI methods have struggled to create formulas for mathematical constants, as such formulas must be valid for an infinite number of digits. “Almost correct” formulas fail to provide insight into the correct ones. A key issue is the lack of a distance metric between a potential formula and the target constant, a metric essential for guiding an automated discovery process.
This study presents a systematic methodology for classifying, characterizing, and identifying patterns in such formulas. The key innovation lies in introducing metrics based on the convergence dynamics of formulas rather than their numerical value. These metrics enable, for the first time, the automated clustering of mathematical formulas. The methodology is demonstrated on continued fraction formulas, which are known for their relationships with mathematical constants and structures.
The research tested this methodology on a dataset of 1,768,900 formulas, successfully identifying numerous known formulas for mathematical constants and discovering previously unknown formulas for constants such as π, the golden ratio, Euler’s constant, and many others. The uncovered patterns allow for the direct generalization of individual formulas to infinite families, revealing rich mathematical structures. This success paves the way for a generative model capable of creating formulas with defined mathematical properties, thereby accelerating the pace of mathematical discoveries.
Amid recent global discussions about whether AI can produce true innovation or merely recombine previous works it has learned from, this work demonstrates genuine mathematical innovation without human intervention.
The research was recently presented at NeurIPS 2024, the world’s largest artificial intelligence conference, which serves as a central platform for showcasing groundbreaking advancements in AI.
One of the fundamental principles of the Ramanujan Group, led by Prof. Ido Kaminer, is the aspiration to make mathematical research accessible to as broad an audience as possible. The group shares its results in an educational format on their website, creates explanatory videos tailored to audiences with limited mathematical backgrounds, collaborates with researchers worldwide, and more. Highlighting this commitment to accessibility, one of the paper’s two lead authors is Uri Seligmann, a high school student who explored the topic under the group’s guidance over the past year and a half as part of the Alpha Program for gifted students.
Another leading contributor to this groundbreaking work is Michael Shalyt, a Research Associate, whose expertise and dedication were instrumental in achieving these remarkable results.
