General-Purpose LLM Solves 80-Year-Old Math Problem in Under 32 Hours for $1,000

The Erdős planar unit distance problem, formulated by mathematician Paul Erdős in the 1940s, asks how many distinct pairs of points in a plane can be exactly one unit apart. OpenAI's LLM disproved the conjecture by constructing a counterexample, a task previously thought to require specialized symbolic reasoning systems. Unlike dedicated solvers such as AlphaProof, the model used here is a general-purpose LLM, suggesting that the reasoning capability generalizes beyond mathematics. As the authors note in their opinion letter, 'This is a disproof, not a proof, which would have been far more challenging.' The 125-page output includes a notable 'page 39 moment,' where the model's reasoning crystallized into a breakthrough.

OpenAI's result was achieved in under 32 hours and for less than $1,000, a fraction of the cost typically associated with solving problems of this complexity. This marks a stark contrast to previous efforts, which often required months of specialized computing resources or human expertise. The affordability of this approach suggests that commodity LLMs could democratize access to advanced problem-solving capabilities. Organizations and individuals can now tackle challenges that were once the exclusive domain of well-funded research institutions. This shift aligns with broader trends in AI, where general-purpose models increasingly commoditize capabilities once requiring specialized systems.

This achievement is not an isolated incident but part of a broader trend. The 2025 International Mathematical Olympiad (IMO) Gold result, also achieved by a general-purpose LLM, demonstrated that these models could excel in domains previously thought to require specialized training. Together, these results suggest that general-purpose agents are becoming increasingly capable of handling complex, open-ended problems. The implications extend beyond mathematics: similar reasoning capabilities could be applied to fields such as physics, biology, and engineering. The ability to solve hard problems without domain-specific training marks a fundamental shift in what AI agents can achieve.

The ability of general-purpose LLMs to tackle complex problems signals a new era for AI agents. Users can now rely on off-the-shelf models for tasks that previously required custom-built systems. This development accelerates the commoditization of advanced problem-solving capabilities, making them accessible to a broader audience. However, it also raises questions about the role of specialized systems and the potential for misuse. As general-purpose agents become more powerful, the challenge will be to harness their capabilities responsibly while ensuring they complement, rather than replace, human expertise.