When Benjamin Grayzel '24 learned that amateur mathematicians were teaming up with AI to tackle open math problems, he wanted to see if the claims added up.
His quest was successful. Grayzel, now a master's student in the Department of Computer Science, used Gemini 3 Pro to find the first complete solution to the 659th problem posed by Paul Erdős.
In a career spanning six decades, Hungarian mathematician Paul Erdős posed more than 1000 mathematical questions, a majority of which remain unsolved.
Now, mathematicians collaborating with AI models such as ChatGPT are beginning to generate solutions to the more obscure problems on the list that are considered relatively easier—they don't make the list of the more high-profile, hard problems that mathematicians want to solve.
Grayzel, who majored in mathematics and government as an undergraduate, picked one such problem—Erdős #659—that he could understand and seemed to him, "likely to have a solution that was more easily verifiable," but that hadn't been tackled extensively by others. The original source of the problem wasn't online, and Grayzel later traced it to the back pages of a book he found at The Dartmouth Libraries.
The problem focuses on distances and points on a flat, two-dimensional plane, and particularly the idea of “distinct distances” within the collection of points, which is the number of different distances between every pair of points.
It asks whether one can find a set of points where the number of distinct distances is more than three but less than a defined upper limit that is a particular function of the total number of points.
Grayzel posed the problem to Gemini 3 Pro, which gave him a solution. After taking it through a few rounds of AI-based verification, he carefully worked through the pieces of the proposed solution that he could understand.
Realizing that there was enough merit in it, he shared the result with Peter Doyle, professor of mathematics and subsequently with Peter Winkler, the William Morrill Professor of Mathematics and Computer Science, who verified that it was indeed, an actual resolution to the problem.
"Sometimes you have to bring an idea from one piece of mathematics, like geometry over to another piece like combinatorics to solve problems and AI is fantastic at that," says Winkler, whose initial response to the news was equal parts delight and caution. "Benjamin chose the problem wisely and was rewarded when it worked."
It's critical to fully understand AI-generated responses and ensure that the math is indeed sound, Winkler says. "AI doesn't really understand logic, and it's one of the things to watch out for," he says.
The key to the solution was the construction of a regular lattice of points that had fewer distinct distances than the upper limit. The construction that defined the set of points that would satisfy the constraints was not new, Grayzel says. It had been proposed before.
However, if the lattice contained one of six shapes—a square, rhombus, equilateral triangle and center, regular pentagon trapezoid, or two instances of an equilateral triangle and point—there would be only two distinct distances, not three or more.
Only five of these had been ruled out previously for the proposed construction. The AI-ideated solution helped close the problem by showing that the sixth case could not fail either.
Grayzel published the solution on ArXiv and the public Erdős problem website, where it quickly caught the attention of Terrence Tao, a celebrated mathematician and professor at the University of California, Los Angeles, who, with other experts, vets and curates the AI-assisted solutions that are submitted.
Tao and other reviewers have since verified the solution, acknowledging the novelty in ruling out the last construction, while noting that the AI may have drawn from previous attempts at solving the problem.
"We now have a fuzzy boundary between what is AI novelty in problem solving, and what counts as help since these tools have access to the entire internet for reviewing existing literature," says Grayzel.
What's exciting, he says, is that the tools could enable "citizen mathematicians" like himself—people who have basic training and interest but not advanced expertise—make real contributions and move the field forward. To reach a wider audience, Grayzel also created a blog post to explain the problem and its solution.
"It was very enterprising of Benjamin to look up a good problem, and to follow through," says Winkler. "AI is an exciting new resource for research and it's great when used well."