GPT-5.4 Pro used von Mangoldt chains to solve several of the most difficult number theory conjectures.
A high-level AI model identified a specific method that human mathematicians overlooked since the problem was first posed in the 20th century. Paul Erdős offered cash prizes for these proofs because he believed they touched on the deepest properties of divisibility and prime numbers. These primitive sets are collections of integers where no number divides another, and their density was a central mystery in number theory for eighty years. The AI suggestion successfully bridged the gap between these sets and the distribution of primes, resolving multiple long-standing conjectures at once. This breakthrough suggests that future mathematical discoveries will increasingly rely on machine-led intuition to solve problems humans find too complex. It marks the first time a major unsolved conjecture has been cracked using a strategy explicitly proposed by an artificial intelligence.
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
arXiv · 2605.00301
A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erdős's seminal 1935 paper.As applications, we prove two 1966 conjectures of Erdős-Sárközy-Szemerédi, on primitive sets of large numbers (#1196) and on divisibilit