AI Models Achieve Breakthroughs in Advanced Mathematics
A recent development in AI has captured attention in the mathematics community after Neel Somani, a software engineer and former quant researcher, tested OpenAI’s latest model. In a surprising turn of events, after inputting a complex mathematical problem into ChatGPT and allowing it to process for 15 minutes, Somani returned to find a complete solution. Upon reviewing the proof with a tool called Harmonic, he confirmed its accuracy.
Somani expressed his curiosity in determining when large language models (LLMs) could effectively tackle open mathematical problems. His findings indicated that the latest AI iteration has made significant advancements in this area.
ChatGPT demonstrated remarkable capabilities by recalling mathematical principles such as Legendre’s formula and Bertrand’s postulate. Interestingly, it also unearthed a 2013 Math Overflow post by Harvard mathematician Noam Elkies, who posed a related solution. However, ChatGPT’s conclusion notably differed, presenting a more comprehensive solution to a problem initially suggested by the celebrated mathematician Paul Erdős, who is renowned for his extensive collection of unsolved problems.
This development raises eyebrows for those skeptical of AI’s role in intellectual pursuits. AI technology is rapidly becoming instrumental in various mathematical domains, from LLMs geared towards formalization to research tools like OpenAI’s deep research capabilities. Somani noted that since the release of GPT 5.2, which he claims excels in mathematical reasoning compared to prior versions, an overwhelming number of problems have been solved, prompting discussions about the potential of AI to advance human knowledge.
Specifically, Somani focused on Erdős problems, a collection of over 1,000 conjectures maintained online. These problems, varying in complexity, have become prime candidates for AI exploration. Notably, a Gemini-powered model named AlphaEvolve initiated a wave of autonomous solutions last November, and more recently, GPT 5.2 has shown remarkable proficiency in high-level mathematics.
Since late December, 15 problems previously classified as “open” on the Erdős website have been marked as “solved,” with AI models credited in 11 of these cases.
Mathematician Terence Tao provided further insights on his GitHub page, documenting eight instances of meaningful autonomous progress made by AI with Erdős problems, alongside six additional cases where significant progress was achieved through existing research. While achieving fully autonomous problem-solving in mathematics remains a distant goal, it is evident that large models have a growing significance in this domain.
On social media platform Mastodon, Tao theorized that the scalability of AI systems positions them uniquely to address the myriad lesser-known Erdős problems, many of which have straightforward solutions.
In addition, a trend toward formalization—an intricate process that enhances mathematical verification—has gained traction. Although formalization can be performed without AI, emerging automated tools like Microsoft Research’s Lean have simplified the process. AI applications, such as Harmonic’s Aristotle, aim to automate much of this labor-intensive task.
According to Harmonic founder Tudor Achim, the surge in solved Erdős problems is noteworthy, but the broader impact lies in the acknowledgment of AI tools by respected mathematicians. “What matters most is that math and computer science professors are now utilizing [AI tools],” Achim commented. “These individuals have their reputations on the line, and their endorsement of platforms like Aristotle and ChatGPT provides compelling evidence of AI’s utility in the field.”
