AI Cracks Erdős Mathematical Problems: How Artificial Intelligence Is Solving Decades-Old Conjectures

The Legacy of a Legendary Mathematician's Problems

Paul Erdős (1913-1996) was one of the most prolific mathematicians of the 20th century, publishing approximately 1,500 papers and collaborating with over 500 mathematicians throughout his career. This Hungarian-born mathematician was known as "the wanderer of mathematics" — he held no permanent academic position, spending his life traveling between universities and research institutions around the world with nothing but a half-empty suitcase. His collaborative network was so vast that the mathematics community invented the "Erdős Number" to measure a scholar's collaborative distance from him: those who co-authored a paper directly with him have an Erdős number of 1, those who co-authored with those scholars have a 2, and so on. This concept later inspired academic network analysis and the application of "six degrees of separation" theory in the scientific community.

Throughout his long academic career, Erdős posed a vast number of unsolved mathematical conjectures. These "Erdős Problems" span multiple core fields including combinatorics, number theory, and graph theory.

These problems share a notable characteristic: they are elegantly stated and easy to understand, yet extraordinarily difficult to solve. Many have stumped mathematicians for decades, serving as important benchmarks for measuring progress in their respective fields. Erdős himself even offered monetary rewards for some problems — ranging from $25 to $10,000 — to incentivize future mathematicians to take them on. The reward amounts directly reflected his assessment of difficulty: problems about the distribution of primes in arithmetic progressions carried rewards of several thousand dollars, while certain combinatorial construction problems might only be worth a few hundred. Even after Erdős's death, a fund managed by Ronald Graham and others continued to honor these rewards. To date, the Erdős Problems website (erdosproblems.com) catalogs over 900 problems, many of which remain unsolved.

Now, with the rapid advancement of artificial intelligence, these legendary problems are being solved at an unprecedented pace, drawing widespread attention from both the mathematics and AI communities.

Why AI Can Crack Mathematical Problems

The Leap in Reasoning Capabilities of Large Language Models

In recent years, AI systems represented by large language models have achieved breakthrough progress in mathematical reasoning capabilities. These models can not only understand complex mathematical language but also perform multi-step logical deductions. In particular, the new generation of models equipped with Chain-of-Thought (CoT) reasoning can decompose complex problems into intermediate steps and solve them progressively, much like human mathematicians.

Chain-of-Thought reasoning was formally proposed in 2022 by Jason Wei and colleagues at Google Research as a prompt engineering technique. Its core idea is to have the model explicitly generate intermediate reasoning steps before arriving at a final answer. Subsequent variants include Tree-of-Thought, which allows models to explore multiple reasoning paths and backtrack; and OpenAI's o1/o3 series models, which incorporate reinforcement learning into the reasoning process, enabling models to engage in longer, deeper internal "thinking." The combination of these techniques has brought AI performance on International Mathematical Olympiad (IMO)-level problems to near gold-medal levels.

What makes Erdős problems particularly well-suited for AI is that many require clever constructions or combinatorial tricks rather than lengthy technical proofs. This aligns perfectly with AI's strengths in pattern recognition and large-scale search — AI can rapidly explore vast solution spaces, trying various possible constructions and finding elegant solutions that humans might overlook.

Deep Integration of Formal Verification and AI

Notably, AI breakthroughs in mathematics are often accompanied by the use of formal verification tools. Proof assistants like Lean can rigorously machine-verify AI-generated proofs, ensuring flawless reasoning.

Lean is an open-source theorem prover initiated by Leonardo de Moura at Microsoft Research in 2013, now in its fourth generation (Lean 4). Its accompanying mathematical library Mathlib contains over 150,000 theorems and definitions, covering mathematical content from undergraduate analysis to the research frontier. The core advantage of formal verification is that it transforms mathematical proofs into code that can be checked line by line by a computer, fundamentally eliminating logical gaps that human reviewers might miss. In 2023, DeepMind's AlphaProof system combined large language models with the Lean formal environment to successfully solve several IMO competition problems, pioneering a new paradigm for AI-driven formal mathematical proofs. Similar formal tools include Coq and Isabelle, which together form the technological ecosystem of modern formal mathematics.

This collaborative model of "AI generation + machine verification" greatly enhances the reliability of mathematical discoveries and makes the mathematics community more willing to accept AI-produced solutions.

Significance and Controversies of AI Solving Mathematical Problems

A Profound Shift in the Mathematical Research Paradigm

AI's success in cracking Erdős problems signals that mathematical research is undergoing a profound paradigm shift. In the past, solving major mathematical problems relied on flashes of individual genius or sustained collective effort; now, AI is becoming a powerful collaborator for mathematicians, and in some cases can independently complete the entire process from proposing ideas to delivering complete proofs.

This shift has sparked deep discussions about the nature of mathematics: Can the beauty and creativity of mathematics be replicated by algorithms? When machines can rapidly produce proofs, how will the role of human mathematicians evolve? These questions have no simple answers, but what is certain is that AI is reshaping how mathematical research is conducted.

The Need for Rational Scrutiny

While news of AI cracking difficult problems is exciting, we must remain clear-headed. First, the problems solved by AI often belong to relatively lower difficulty levels or have clear solution paths — the true "problems of the century" remain far out of reach. The mathematical problems AI can currently tackle typically share these characteristics: solutions of finite length, obtainable through combinatorial search or constructive methods, or natural extensions of existing technical frameworks. By contrast, millennium problems like the Riemann Hypothesis, P≠NP, or the smoothness of Navier-Stokes equations often require the creation of entirely new mathematical theoretical frameworks — something current AI cannot yet accomplish. Furthermore, many frontier mathematical breakthroughs still depend on uniquely human intuition — the ability to connect seemingly unrelated mathematical fields — and this kind of cross-domain conceptual innovation remains a weak point for AI.

Second, AI-produced solutions must undergo rigorous dual verification by both humans and machines before being formally recognized by the mathematics community.

Additionally, questions about "AI originality" deserve serious consideration: Is AI truly "creating" new mathematics, or is it retrieving and recombining existing knowledge from vast training data? This debate is unlikely to be resolved in the near term, but it reminds us that while celebrating AI achievements, we should also carefully assess its genuine contributions.

Outlook: A Future of Human-AI Collaboration in Mathematics

The successive solutions to Erdős problems are merely the beginning of AI's entry into the halls of mathematics. It is foreseeable that as model capabilities continue to improve and formal tools become increasingly mature, artificial intelligence will demonstrate its power across more mathematical domains. Deep collaboration between human mathematicians and AI may well become the new norm for unlocking deeper mathematical mysteries.

Regardless, the intellectual legacy left by this legendary mathematician is being continued and advanced in ways he himself might never have imagined. And this mathematical revolution of human-AI collaboration has only just begun.

Key Takeaways