Summary of Key Points
80 years ago, mathematician Erdős posed a seemingly simple question: If n points are placed on a plane, what is the maximum number of pairs of points that are exactly 1 centimeter apart? He speculated that the growth rate of this number would be close to n (almost linear, such as n multiplied by a slowly decreasing factor) and would not exceed an “almost linear” range like n^1.01. In May of this year, OpenAI’s general-purpose AI model, which was not specifically trained for mathematics, used complex algebraic and number theory methods to refute this conjecture—proving that there exists a small positive number δ (such as 0.014) such that the number of pairs of points at a unit distance can be as high as n^(1+δ), indicating a faster growth rate than Erdős predicted. This marks the first time that AI has made a breakthrough in a major mathematical conjecture. The proof, led by Chen Lijie, a legend from competitive mathematics, was presented in a format understandable to humans, sparking discussions about whether AI could revolutionize mathematical research.
I. Erdős’ Conjecture: What is the Maximum Number of Points at a “1-Centimeter Distance” on a Plane?
Imagine drawing n points on a piece of paper and trying to arrange them so that as many pairs of points as possible are exactly 1 centimeter apart. Erdős’ question was: What is this maximum number?
- The simplest case: A single central point with n-1 points located on circles with a radius of 1 centimeter would result in n-1 pairs at a unit distance (from the center to each point).
- But what about more complex arrangements? For example, on a grid of integers, each point could have several neighbors at a distance of 1 centimeter, but plane geometry limits this to a maximum of three points that can be equidistant from each other (in an equilateral triangle), so it’s not possible for all pairs to be 1 centimeter apart.
Erdős conjectured that the growth rate would be n^(1+o(1)), meaning it would be slightly faster than n but not significantly so (not as explosive as n^2). This conjecture has puzzled mathematicians for 80 years.
II. AI’s Innovative Approach: Using Higher-Dimensional Mathematics to Overcome Plane Constraints
While humans have previously tried using two-dimensional grids (such as chessboards) to construct the maximum number of pairs at a unit distance, AI took a path that few had explored:
1. Moving to Higher-Dimensional Number Fields: Humans used ordinary integers (in 2 dimensions), while AI utilized more complex “higher-dimensional number systems.” In these systems, a number can be factored into more components, similar to cutting an apple into smaller pieces to create more pairs of points at the same distance.
2. Balancing Algebra and Geometry: Points in higher dimensions would cluster together when projected onto a 2D plane (like pressing a 3D wire into a wall, resulting in many intersections). AI used “infinite expansion number field towers” (combined with Goldbach-Scharafovich theorem) to ensure that enough pairs at a unit distance could be generated while controlling the density in the 2D projection.
The result: AI proved that the number of pairs at a unit distance can be as high as n^(1+0.014), directly disproving Erdős’ conjecture.
III. The Milestone Significance: AI’s First Major Mathematical Conquest
The significance of this achievement lies in:
- It’s Not a Specialized Tool: AI is a general-purpose model, not trained specifically for mathematics, and it did not use any special hints or tools.
- Top-Level Recognition: Fields Medalist Terence Tao stated that if this AI proof were written by a human, it could be published in the prestigious mathematical journal *Annals of Mathematics* (the same journal where Fermat’s Last Theorem was proven).
- Breaking Through Intuitive Barriers: Mathematical conjectures require creative thinking to devise new methods. It was previously believed that this ability was unique to humans, but AI has demonstrated that it can not only solve problems but also generate new proof approaches.
IV. The Key Role of Chen Lijie: A Legend from Competitive Mathematics
Behind this breakthrough is a Chinese mathematician: Chen Lijie.
- A Competition Legend: He won the gold medal in informatics at 16 and the IOI (International Olympiad in Informatics) at 18, graduated from Tsinghua University, and holds a Ph.D. from MIT.
- The Bridge Between AI and Mathematics: He joined OpenAI earlier this year and led the effort to transform the AI’s proof into a format understandable to humans (the original AI reasoning process was 125 pages long). He said, “I didn’t expect such a major breakthrough in just five months.”
Chen Lijie’s role highlights that AI’s findings require human experts to interpret and verify them, indicating that collaboration between humans and AI is a crucial model for current mathematical advancements.
V. AI vs. Human Mathematics: Assistant or Replacement?
This breakthrough has sparked intense debate:
- AI’s Advantages: It can handle high-dimensional mathematics beyond human imagination, has a low cost of trial and error, and excels at combining existing tools. Some even argue that the human brain may not be the optimal configuration for advanced mathematics.
- AI’s Limitations: AI cannot propose new research directions or create new theories (like Erdős did with his conjectures); it can only combine existing mathematical tools.
- Reflecting the Moravec Paradox: Mathematics is difficult for humans but easy for AI; however, tasks like cooking and walking, which are simple for humans, are challenging for AI.
The current consensus is that AI is a powerful assistant, but it has not yet replaced human mathematicians. Human abilities such as “taste,” “curiosity,” and the ability to pose new questions remain irreplaceable by AI.
Conclusion
AI’s refutation of Erdős’ conjecture marks a starting point rather than an end. It forces us to reconsider whether mathematics is truly the last bastion of human intelligence. In the future, collaboration between AI and human mathematicians could lead to even more groundbreaking discoveries. For now, humans remain the “soul” of mathematical research—AI handles the “calculation and construction,” while humans provide the “direction and creativity.”