AI Tools Revolutionize Solutions for Old Math Problems Andreser/Getty Images
Amateur mathematicians are leveraging artificial intelligence chatbots to tackle historic mathematical challenges, much to the astonishment of experts. Although the questions may not represent the pinnacle of mathematical complexity, their successful resolution suggests a significant breakthrough in AI’s capabilities in mathematics, potentially altering future methodologies, according to researchers.
The challenges addressed by AI are linked to Paul Erdős, a renowned Hungarian mathematician celebrated for posing intriguing yet complex questions throughout his prolific 60-year career. “The inquiries were often straightforward but exceedingly complex,” says Thomas Bloom from the University of Manchester, UK.
At the time of Erdős’ death in 1996, over 1,000 unsolved problems existed, spanning various mathematical disciplines, from combinatorics to number theory. Today, these challenges represent critical milestones for advancements in mathematics, Bloom explains. He maintains a website dedicated to cataloging these problems and tracking mathematicians’ progress in solving them.
Given the clarity of Erdős’ problems, mathematicians began experimenting with feeding them into AI tools like ChatGPT. Last October, Bloom noted an increase in users employing AI models to uncover pertinent references in mathematical literature to aid their solutions.
Shortly thereafter, AI tools began uncovering partial improvements in results—some were previously documented while others seemed to be novel.
“I was taken aback,” Bloom recalls. “Previously, when I tested ChatGPT, it provided mere conjectures, leading me to abandon it. However, since October, I discovered genuine papers, as ChatGPT effectively analyzed existing literature, uncovering substantial insights.”
Inspired by these advancements, Kevin Barrett, an undergraduate mathematics student at Cambridge, along with amateur mathematician Liam Price, set out to identify simpler and less-explored Erdős problems amenable to AI solutions. After discovering the number 728—a conjecture in number theory—they successfully solved it using ChatGPT-5.2 Pro.
“Upon seeing the statement, I thought, ‘Perhaps ChatGPT can solve this. Let’s give it a shot,’” Barrett remarks. “Indeed, numerous experts concur that the argument is elegant and quite sophisticated.”
After ChatGPT generated the proof, Barrett and Price employed another AI tool named Aristotle, developed by Harmonic, to validate their findings. Aristotle translates traditional proofs into the Lean mathematical programming language, which is swiftly verified for accuracy by a computer. Bloom highlights this process as vital, as it conserves researchers’ limited time when confirming their results’ validity.
As of mid-January, AI tools have completely solved six Erdős problems, but professional mathematicians later identified that five of these had existing solutions in the literature. Only problem number 205 was entirely resolved by Barrett and Price without prior solutions. Additionally, AI facilitated minor improvements and partial resolutions to seven other problems that were absent in existing literature.
This predicament has sparked debate regarding whether these AI tools unveil true innovations or simply resurrect old, overlooked solutions. Bloom notes that AI models frequently need to reconceptualize problems, discovering papers that make no mention of Erdős whatsoever. “Many papers I encountered would likely have remained undiscovered without this kind of AI documentation,” he remarks.
Another point of discussion is the potential limits of this approach. While the addressed problems aren’t the most formidable in mathematics, they could typically be resolved by first-year doctoral students; nonetheless, Bloom considers the achievement significant, noting the substantial effort required for such tasks.
Barrett further emphasizes that the problems currently being solved are relatively easier compared to more challenging Erdős problems, which contemporary AI models struggle to tackle. “Ultimately, AI will need more advanced models to address complex problems,” he forecasts. Some of these challenging issues even come with cash prizes for solutions, although Barrett believes that resolutions are unlikely in the near future, stating, “I don’t think we have a model for that yet.”
Utilizing AI to tackle Erdős’ problems offers promising potential for progress, according to Kevin Buzzard. Since most of the addressed challenges are straightforward or have received scant attention, it’s difficult to gauge whether these results signify substantial breakthroughs or if they warrant professional concern. “This is progress, but mathematicians aren’t quite ready to embrace it fully,” Buzzard observes. “It’s merely a budding advancement.”
Even with the models’ current limitations, their capability to work with moderately complex mathematics could fundamentally transform how researchers craft and analyze proofs. This advancement allows mathematicians with specialized knowledge to access insights from diverse mathematical fields.
“Few individuals possess expertise across all mathematical domains, limiting their toolkit,” Bloom explains. “Being able to obtain answers rapidly, without the hassle of consulting others or investing months in potentially irrelevant knowledge, creates numerous new connections. This is a groundbreaking shift that is likely to widen the scope of ongoing research.”
It may enable mathematicians to adopt entirely novel methodologies. Terence Tao at the University of California, Los Angeles, has been instrumental in validating AI-assisted methods for solving Erdős problems.
Given their limited schedules, mathematicians often prioritize a select few difficult problems, leaving many easier yet essential questions overlooked. If AI tools can be employed instantaneously across a multitude of problems, Tao believes it could facilitate a more empirical approach to mathematics, enabling extensive testing of various solutions.
“Currently, we neglect 99% of solvable problems due to our finite resources for expert analysis,” Tao asserts. “Therefore, we often bypass hundreds of significant issues, seeking just one or two that capture our interest. We also lack the capacity for comparative studies like, ‘Which of these two methods is superior?'”
“Such large-scale mathematics has yet to be undertaken,” he concludes. “However, AI demonstrates the feasibility of this approach.”
Topics:
- Artificial Intelligence/
- ChatGPT
Source: www.newscientist.com
