OpenAI AI model disproves 80-year-old Erdős unit distance conjecture
The breakthrough marks the first autonomous proof of a major open conjecture, though human verification remains essential to validate the findings.
In mid-May 2026, OpenAI announced that an internal artificial intelligence model had disproved the Erdős unit distance conjecture, a problem in discrete geometry that had remained unsolved for eight decades. The model utilised techniques from algebraic number theory to demonstrate that the maximum number of unit-distance pairs grows faster than previously conjectured by Paul Erdős. While the result does not fully resolve the problem, it marks the first instance of an AI autonomously producing a proof for a major open conjecture. Human mathematicians, including Fields Medalist Tim Gowers, verified the result and noted the collaboration between AI’s computational breadth and human insight.
The Erdős unit distance problem, introduced in 1946, asks for the maximum number of pairs of points at unit distance apart within a set of n points. Erdős conjectured a lower bound for this growth rate based on grid structures, which remained the best-known lower bound for 80 years. OpenAI’s AI model constructed a grid in a high-dimensional space and projected this structure into two dimensions, using algebraic integers to build a more complicated grid than Erdős’s original square grid. This new pattern allowed for more unit-distance pairs, with human mathematician Will Sawin showing the growth rate is at least n^1.014, whereas Erdős’s lower bound was n^(1 + C/(log log n)).
The AI’s solution relied on applying sophisticated techniques from algebraic number theory, a field with which many human mathematicians working on the unit distance problem were not simultaneously engaged. The AI model solved the problem only half of the time even with the maximum token budget, highlighting the "grind" of trying many proof strategies that do not work. Other AI systems have also made progress: GPT-5.2 and Harmonic’s Aristotle produced an autonomous solution to an Erdős problem in January 2026, and Google announced in May 2026 that its AI solved nine open Erdős problems. University of Michigan postdoc Xiao Ma found that GPT-5.5 could also prove Erdős wrong if given a small hint.
The problem remains partially unresolved, as the best upper bound for unit distances is around n^1.333, leaving a gap to be closed. The achievement highlights a new paradigm where AI systems leverage broader knowledge of past work and willingness to explore tedious strategies, while humans provide insight and verification. This dynamic suggests a medium-term future where human mathematicians and AI models complement each other, with AIs tackling curated lists of problems and humans focusing on deeper conceptual questions.
The announcement has been met with significant interest from the mathematical community. Tim Gowers, who won the Fields Medal, described the solution as a milestone in AI mathematics. Daniel Litt, a University of Toronto professor, noted that this is the first example of a result produced autonomously by an AI that is exciting in itself. The event underscores the rapid advancement of AI capabilities in mathematics, moving from struggling with basic arithmetic to resolving complex, long-standing conjectures.
