A 23-year-old amateur with no advanced mathematical training has reportedly used ChatGPT to solve a problem that has stumped world-class mathematicians for six decades. The breakthrough, achieved by Liam Price, marks a potential turning point in how artificial intelligence interacts with complex mathematics.
The Breakthrough: Beyond “Pattern Matching”
While AI has recently been credited with solving several “Erdős problems”—conjectures left behind by the legendary mathematician Paul Erdős—experts have remained skeptical. Many previous AI successes were criticized for being unoriginal, essentially re-stating known truths or following established paths.
Price’s discovery appears fundamentally different. By prompting a high-level LLM (Large Language Model), he elicited a solution that bypassed the “mental block” that had stalled human experts for years.
“The LLM took an entirely different route,” says Terence Tao, a prominent mathematician at UCLA. “There was a standard sequence of moves that everyone who worked on the problem previously started by doing… the LLM used a formula that was well known in related parts of math, but which no one had thought to apply to this type of question.”
The Problem: Primitive Sets and the “Score” of Numbers
To understand the significance, one must look at the nature of the problem. The challenge concerns primitive sets —collections of whole numbers where no single number in the set can be evenly divided by another.
The mathematical context is as follows:
– Prime Numbers as a Foundation: A prime number is only divisible by itself and one. A “primitive set” essentially generalizes this concept to an entire group of numbers.
– The Erdős Sum: Erdős developed a way to calculate a “score” (a sum) for these sets.
– The Conjecture: Erdős hypothesized that as the numbers in a primitive set grow toward infinity, the “score” would approach a limit of exactly one.
For decades, mathematicians—including Stanford’s Jared Lichtman—attempted to prove this limit, but they all hit the same wall. The problem wasn’t that the math was impossible; it was that human intuition kept leading researchers down the same incorrect paths.
“Vibe-Mathing”: A New Approach to Discovery
The method used by Price and his collaborator, Kevin Barreto, has been jokingly dubbed “vibe-mathing.” Rather than approaching the problem with rigorous, traditional proofs, they used AI to explore open problems at random, testing the “vibe” of the AI’s mathematical reasoning.
However, the process is not as simple as clicking a button. Experts note several critical caveats:
1. Low-Quality Raw Output: The initial proof generated by ChatGPT was “quite poor” in its presentation. It required human experts to sift through the text to find the underlying logic.
2. Human-AI Collaboration: The breakthrough was not the AI working in isolation, but the AI providing a “cognitive leap” that humans then refined, distilled, and validated.
3. A New Tool for Anatomy: Mathematicians like Tao and Lichtman suggest this isn’t just about solving one old puzzle; it’s about a new way to understand the “anatomy” of large numbers.
Why This Matters for the Future of Science
This event raises a profound question: Is AI capable of genuine mathematical intuition?
If an LLM can identify a connection between two seemingly unrelated mathematical fields—something humans missed due to cognitive bias—it suggests that AI might serve as a “lateral thinker” in scientific research. While the long-term significance is still being debated, the ability of AI to break through long-standing human mental blocks suggests it may move from being a mere calculator to a genuine collaborator in discovery.
Conclusion: By utilizing an unconventional mathematical connection that humans had overlooked, an amateur using AI has provided a new roadmap for solving complex number theory problems, signaling a shift in how we might approach scientific breakthroughs.
