Erdős problem

Paul Erdős was a Hungarian mathematician who lived from 1913 to 1996. He owned almost no possessions, had no fixed home, and traveled from university to university his entire life. Above all, though, he asked questions: hundreds of mathematical puzzles that he could not solve himself. For many of them he offered his own prize money, usually between 25 and several thousand dollars. These questions are known today as Erdős problems. What makes them special is the contrast between form and difficulty: they can often be explained in two sentences, yet some have remained open for over sixty years.

Why Erdős' questions became a benchmark

In mathematics, a question is only considered answered once there is a complete, gapless proof. A proof is a chain of arguments in which each step necessarily follows from the previous one. A computer can check millions of examples and find no counterexample – yet nothing is proven by that. This is exactly why Erdős problems are so stubborn. They cannot be settled by mere trial and error.

For research, they are also productive. Anyone who attacks one of these problems often has to invent new methods. These methods then help with entirely different questions. The path to the solution is thus frequently more valuable than the solution itself. Erdős knew this and chose his questions deliberately with that in mind.

Since around 2024, the problems have taken on a second role. AI companies use them as a touchstone for their language models, i.e., for programs like ChatGPT that generate text. The reason is simple: these tasks appear in no textbook with a ready-made solution. A model cannot retrieve them from memory. If it makes progress here, it has genuinely accomplished something.

What such a question specifically demands

Most Erdős problems come from number theory or combinatorics. Number theory deals with the properties of whole numbers, combinatorics with the question of how many ways things can be arranged. A typical pattern goes: if a set of numbers is large enough, a certain pattern must inevitably appear within it. What must then be shown is at what size this holds true – and why there can be no exception.

A well-known example is the conjecture on arithmetic progressions. Erdős asked: if you choose an infinite set of numbers such that the sum of their reciprocals becomes infinitely large, does it then always contain arbitrarily long sequences of numbers with equal spacing? Part of this question was solved in 2004; the rest remains open to this day. Erdős had offered $5,000 for it, the highest amount he ever set.

Anyone working on such a problem usually spends months on it. One searches for special cases, sorts them into groups, and tries to rule out each group individually. Often one gets stuck on a single case that refuses to fit. This is precisely where AI systems now come into play: they propose approaches that a human then checks.

The dispute over the AI solutions

The British mathematician Thomas Bloom runs a website where he collects around 1,000 Erdős problems and documents their status. This list has become the reference that everyone relies on. When a problem there is marked as “solved,” it is a small piece of news in the field.

In October 2025, an incident caused a stir. OpenAI employees publicly reported that their model had cracked several open Erdős problems. Bloom clearly disagreed. The model had not found any new proofs but had tracked down old academic papers in which the solutions had long since been published. Bloom simply hadn’t known about them. The posts were withdrawn, and the debate continued for days.

The case illustrates an important distinction. Rediscovering forgotten literature is useful work, but it is not new mathematics. In the meantime, however, there have also been genuine advances: on individual problems, models have contributed proof steps that were subsequently reviewed and accepted by experts. So if you read in the news that an AI has solved an Erdős problem, it’s worth asking – did it find something, or did it invent something?