Erdős Problems: The Role of AI in Solving Mathematical Conjectures
One idle evening last October, Mehtaab Sawhney, a mathematician at Columbia University, found himself perusing the website erdosproblems.com. This site is an updated record of the 1,179 conjectures left behind by the prolific mathematician Paul Erdős. Sawhney had always been intrigued by Erdős problems, which range from minor curiosities to central open problems in number theory and combinatorics.
As Sawhney delved into the problems listed on the website, he stumbled upon #339, a problem that seemed deceptively simple for it to still be classified as “open” nearly two decades after Erdős’s passing. He had encountered similar conjectures in the past, and typically turned to Google for answers. However, on this occasion, he decided to use ChatGPT, a language model, to assist him in his search. To his surprise, ChatGPT was able to locate a reference to a solution for the problem.
Encouraged by this success, Sawhney enlisted the help of fellow mathematician Mark Sellke, who was working for OpenAI at the time. Together, they employed ChatGPT to uncover lost solutions to nine other Erdős problems, as well as partial solutions to 11 additional problems. This collaboration marked the beginning of a surge in activity on the website, with AI tools playing a significant role in transferring approximately 100 Erdős problems into the “solved” category since October.
The emergence of Language Model Models (LLMs) has revolutionized the field of mathematics by providing unprecedented access to vast amounts of literature and aiding mathematicians in formulating new solutions. LLMs have proven invaluable in synthesizing information, guiding researchers towards novel approaches, and even constructing original proofs for previously unsolved problems.
Despite the advancements made possible by AI, there is still a long way to go before machines can fully replace human mathematicians. While LLMs have shown remarkable potential in assisting with mathematical research, major open problems in mathematics remain beyond their current capabilities. However, the integration of AI into the field is inevitable, and mathematicians are beginning to realize the benefits of utilizing LLMs as research assistants.
The story of Erdős problems serves as a testament to the evolving relationship between AI and mathematics. These problems have become a benchmark for testing the capabilities of LLMs, showcasing their ability to navigate complex mathematical landscapes and propose innovative solutions. As the collaboration between mathematicians and AI continues to flourish, it is evident that 2026 may herald a pivotal year where AI-contributed proofs make their way into major mathematics journals.
In conclusion, the intersection of AI and mathematics represents a paradigm shift in the way mathematical research is conducted. As researchers like Mehtaab Sawhney and Mark Sellke embrace this new era of collaboration, the boundaries of mathematical exploration are being pushed further than ever before. With AI as a powerful ally, the future of mathematics looks promising, with endless possibilities waiting to be uncovered.
