Result 011 in OpenAI’s math release is about how p − 1 breaks into primes. Factor a large random number and measure each prime factor by its logarithm: the factors split the number into shares that add up to one, and in 1972 Patrick Billingsley proved that these shares follow a precise limiting law, the Poisson–Dirichlet law PD(1), which also describes a randomly broken stick and the relative cycle lengths of a large random shuffle. For p − 1, where p is a random prime, the same law was widely expected (Ford, Konyagin and Luca wrote it down in 2010) but out of reach, because a large prime factor q of p − 1 puts p in an arithmetic progression with a huge modulus. Three preprints claim to settle it: the full PD(1) law for p − 1; many primes with p − 1 as smooth as you like, which proves Erdős’s conjecture that some values of Euler’s totient function φ are hit more than n^(1 − ε) times; and infinitely many primes with μ(p − 1) = 1. This episode breaks a stick, compares the law with data for primes up to ten million, shows why φ(m) = 72 has seventeen solutions, and sketches the claimed proof.
Where this claim stands (as of October 11, 2026):
∎ Posted as preprints by OpenAI (dated September 17 and 24, released October 6, 2026; 187 pages across three papers)
□ Not yet peer reviewed
□ Not checked in Lean (no Lean proof was published)
It has not been withdrawn. The law is a limit with no rate, and convergence in the data is slow. Erdős problem #821 (the totient question) is still listed as open on erdosproblems.com.
Chapters
0:00 Breaking a number
0:57 A broken stick
1:37 Primes minus one
2:56 The claims
3:40 Totient collisions
4:45 How the proof goes
5:33 The caveats
Read the first paper in OpenAI’s catalog, on smooth shifted primes and totient fibers (the episode page links all three):
Episode page, sources and full transcript:
We checked the numbers in the video with a short Python script: the factorization of 1,000,002, the Dickman values, a simulated broken stick, how often the largest prime factor of p − 1 exceeds its square root for primes up to 10^7, and the seventeen solutions of φ(m) = 72.
How this was made: the script and animations were made with AI assistance (Claude) using the Manim animation library, and the narration is an AI-generated voice. Spot a mistake? Tell us in the comments or open an issue:
AI Proved What? is independent and not affiliated with OpenAI. Source material from github.com/openai/math, used under the Apache License 2.0.
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