Definition
Plain language
A famous open math problem posed by Paul Erdős, often easy to state but very hard to prove.
As stated in the literature
A problem from Paul Erdős's published lists of open conjectures, frequently in combinatorics or number theory; used as a benchmark for AI-driven formal proof search systems.
Also called: Erdős problems
Why it matters: These problems are deep, well-defined, and unsolved, making them a credible yardstick for whether an AI proof system is doing real mathematics.
For example, one Erdős problem asks whether every set of integers with positive density contains three numbers in arithmetic progression — a sentence anyone can read but mathematicians spent decades on.
Heard on the show
“Three hundred and fifty-three open Erdős problems.”Episode 067 — An AI Just Solved a 1996 Erdős Problem—and the Simplest Agent Won