OpenAI solved Paul Erdős's problem that eluded mathematicians for 80 years

OpenAI announced the solution to one of the most famous open problems in mathematics — Paul Erdős's unit distance problem. This problem has puzzled the world's best mathematicians for 80 years, since its formulation in 1946. According to the company, the solution demonstrates that modern AI systems are capable of performing original mathematical reasoning and generating new proofs.

What Is This Problem

The problem belongs to the field of discrete geometry. It was formulated by Hungarian mathematician Paul Erdős — one of the most prolific mathematicians of the 20th century. The problem sounds deceptively simple: imagine that many points are placed on a plane, and any two points are exactly one unit of distance apart.

The question: what is the maximum number of such points that can exist? Despite the simplicity of its formulation, this problem turned out to be one of the most stubborn open problems in mathematics. The world's best scientists have tried to solve it.

Each new improvement in the estimate of the number of points required years of intensive research, and often the development of entirely new mathematical methods. Over eight decades, the problem became legendary in the mathematical community — a symbol of how a seemingly simple problem can hide incredible depth.

How the Neural Network Tackled the Problem

OpenAI used its advanced models with enhanced logical reasoning capabilities. The key point: the model did not simply iterate through known mathematical facts and did not apply standard textbook methods. The system independently formulated a new hypothesis and conducted original proof that was previously considered beyond the capabilities of neural networks. The company emphasizes that this is not the first time AI has demonstrated such abilities. But each new example expands the boundaries of what modern models are capable of in the realm of abstract thinking:

• Generation of original mathematical proofs
• Work with formal logic and verification of reasoning correctness
• Creation of new approaches to long-known problems
• Independent verification of its own logical chains
• Solution of problems requiring deep abstract analysis

Why This Matters for Science

At the academic level, this means that one of the legendary open problems that remained unsolved for eight decades has finally been resolved. At the level of AI development, the achievement demonstrates a fundamental shift: modern neural networks are now capable of performing substantive mathematical reasoning, which was previously considered the exclusive domain of human intelligence. For researchers in the field of AI, this is an important signal that models are becoming increasingly capable of abstract logical thinking. This paves the way for AI application in other complex fields requiring original mathematical thinking — from theoretical physics and cryptography to economics and biochemistry.

What This Means for the Future

Solving the 80-year-old Erdős problem is a milestone in the evolution of AI. It transitions from simple processing of text patterns and known methods to the execution of formal and original mathematical thinking. AI no longer merely applies known solutions, but is capable of generating new ideas and conducting original proofs. Such breakthroughs outline the boundaries of what will be possible to delegate to artificial intelligence in scientific research in the coming years and decades.