Summary of Key Points
In 2026, three AI companies—OpenAI, Anthropic, and DeepMind—made groundbreaking advancements in the field of mathematics: OpenAI disproved the “Unit Distance Conjecture” that had stood for 80 years; Anthropic found a counterexample to the “Jacobi Conjecture,” one of the most challenging problems in 21st-century mathematics; and DeepMind solved nine Erdős problems, including two that had been unsolved for 56 years. These achievements are more independent and significant than previous ones, but they have also raised concerns within the mathematical community about the potential impact of AI on the peer review process, the attribution of research findings, and the direction of scientific inquiry. Mathematicians issued the “Leiden Manifesto,” calling for the regulation of AI’s use in mathematics to preserve the right to decide what problems are worthy of study.
Detailed Explanation
1. What Has AI Achieved This Year? From “Silver Medalist” to “Problem Solver”
2026 has been a year of breakthroughs for AI in mathematics:
- OpenAI: In May, it disproved the “Unit Distance Conjecture,” which Erdős posed in 1946 (how many pairs of points on a plane have a distance of 1? OpenAI’s solution was considered to be at the level of top-tier mathematical proofs.)
- Anthropic: In July, using Fable 5, it found a three-dimensional counterexample to the “Jacobi Conjecture,” one of the 18 most difficult problems in mathematics. Dr. Zhang Yitang had studied this conjecture in his thesis but had to work on other projects due to difficulties with its proof. Anthropic’s counterexample was concise and quickly verified.
- DeepMind: In May, AlphaProof Nexus solved nine Erdős problems, two of which had been unsolved for 56 years. DeepMind was also able to independently verify the correctness of these proofs using the Lean programming language, with costs amounting to just a few hundred dollars.
In contrast, in 2024, AI won only a silver medal in the IMO (International Mathematical Olympiad) by solving problems for which answers were already known; in 2025, GPT-5 claimed to have solved an Erdős problem, but it turned out to be merely a case of extensive literature review. This year’s achievements are truly original and independent, without relying on human-provided solutions or research.
2. Why Are These Achievements So Groundbreaking?
There are two main reasons for the significance of these breakthroughs:
- The Challenges Are Significant: These are not trivial problems but recognized as major hurdles in mathematics. For example, the Unit Distance Conjecture was personally posed by Erdős, and the Jacobi Conjecture was identified by Fields Medalist Smale as a century-long challenge that had seen multiple attempts by mathematicians.
- AI’s Increasing Independence: Previously, AI could generate candidate solutions, but mathematicians were needed to refine them. This year, OpenAI’s proof was entirely autonomous, and DeepMind used the rigorous mathematical language Lean to verify each step of the proof automatically, correcting errors until it was successful—almost as if AI was grading its own work.
Anthropic’s counterexample is particularly remarkable for its simplicity: it can be understood by anyone, which has attracted widespread attention.
3. Is AI More Advanced Than Human Mathematicians?
While AI has clear advantages, it has not yet surpassed humans:
- Cross-disciplinary Ability: Modern mathematics is highly specialized, but AI can integrate methods from different fields. For instance, OpenAI used techniques from number theory to solve the Unit Distance Conjecture, a combination that would be unlikely for a human mathematician.
- Lack of Bias: Humans tend to assume that Erdős’ conjectures are correct, while AI can boldly search for counterexamples without preconceived notions.
- Endless Patience: Finding counterexamples often requires extensive trials, something AI can do tirelessly.
However, AI has a critical weakness: it lacks the ability to identify new research directions. As Associate Professor Li Xinyi from Peking University explained, human mathematical knowledge is like a polygon with uneven edges; top mathematicians can explore unknown areas at the sharp corners, while AI can only fill in the gaps but cannot create new ones. For example, AI cannot address the 23 problems posed by Hilbert.
4. Concerns Within the Mathematical Community
The rapid progress of AI has raised three major concerns:
- Peer Review Overload: AI can generate proofs in hours, while human reviewers take weeks or months to evaluate them. There are few experts capable of assessing such complex work, and a surge in AI-generated proofs could overwhelm the review process, leading to the publication of incorrect results that mislead the public.
- Attribution of Credits: AI relies on existing research, raising questions about who should be credited for the findings. This also raises copyright issues.
- Distortion of Research Focus: Tech companies often focus on problems that are easy to solve and understandable to the general public, rather than those with profound significance but greater difficulty. With limited university funding, mathematicians may collaborate with companies, potentially sacrificing important research. The value of mathematics goes beyond solving problems; it also involves determining what problems are worth studying.
Mathematicians are striving to protect not just their own work but the essence of mathematical research—the right to decide what problems deserve attention.
5. The Future: Will AI Be a Tool or a Competitor?
For now, AI is more like a powerful assistant that helps humans quickly test hypotheses and explore different approaches. However, it still relies on human judgment. For example, OpenAI’s proof required review by nine top mathematicians, and DeepMind’s Lean verification needed human confirmation of its significance.
If AI ever develops the ability to identify new research directions (such as proposing problems similar to Hilbert’s 23), that would mark a true breakthrough. The goal is to regulate the use of AI so that it serves as a tool to enhance mathematical research, not to replace mathematicians.
In summary, AI is transforming how mathematics is conducted, but the core values of understanding, exploration, and judgment remain in the hands of humans. In the future, AI and human mathematicians may work together, with AI facilitating groundbreaking discoveries while humans ensure that research focuses on meaningful and important questions.