ИИ решает задачи Эрдёша: почему легендарные проблемы математики поддаются нейросетям

Neural networks have achieved their greatest mathematical successes precisely on problems posed by Paul Erdős — the iconoclastic Hungarian mathematician of the mid-twentieth century who left behind hundreds of unsolved problems. According to Quanta Magazine dated August 3, 2026, researchers are now examining in detail what makes Erdős problems attractive to AI — in order to understand how neural networks will change the rest of mathematics.

Who Was Erdős and Why Are His Problems Famous

Paul Erdős (1913–1996) was one of the most prolific mathematicians in history: over 1,500 scientific papers co-authored with more than 500 colleagues, and thousands of open problems he formulated. Erdős led a nomadic lifestyle, moving from university to university, and offered a cash prize for each solution — ranging from $10 to $10,000 depending on difficulty. Many problems remained unsolved well into the twenty-first century.

Erdős problems share several distinctive characteristics:

• Concise formulation: most problems can be stated in 2–3 sentences
• Combinatorial nature: primarily number theory and graph theory
• A concrete, verifiable answer: a construction either exists or it does not
• "Elegant difficulty": a simple statement conceals a deep proof

This combination, according to mathematicians interviewed by Quanta Magazine, is precisely what makes Erdős problems amenable to modern AI.

Why Do Erdős Problems Yield to Neural Networks?

The main reason is that the structure of Erdős problems aligns well with how formal proof systems work. Verifiers such as Lean and Coq allow AI to propose a hypothesis and check each statement for correctness step by step. Without such a mechanism, neural networks can generate convincingly worded but mathematically incorrect proofs — a limitation that long restricted the use of AI in science.

Combinatorial problems with a clear yes/no criterion are especially convenient: the model cycles through strategies, constructs examples, and searches for counterexamples without the risk of accumulating errors in intermediate steps. In 2024, Google DeepMind's AlphaProof system solved four out of six problems at the International Mathematical Olympiad (IMO 2024) — several of them were of the combinatorial type characteristic of the Erdős tradition.

What Will This Change in Mathematics?

According to researchers quoted by Quanta Magazine, Erdős problems have become a litmus test for all of mathematics. By figuring out exactly why this class of problems yields to neural networks, scientists will be able to draw a map: where AI will assist mathematicians in the coming years, and where human intuition will remain irreplaceable.

According to

Quanta Magazine's analysis, by studying Erdős problems, mathematicians are effectively seeking an answer to a more general question: what properties make a mathematical problem machine-solvable in principle.

An open question remains whether the same approach works in less formalized fields: algebraic geometry, topology, problems that require the creation of fundamentally new concepts. There, a researcher's intuition remains indispensable for now, and mathematicians emphasize this.

What This Means

AI's successes on Erdős problems are a signal of a qualitative shift in mathematics: neural networks are moving from computational assistance to closing problems that have stood open for decades. Understanding exactly where the boundary of AI's capabilities in science lies is one of the key research challenges for the years ahead.