Erdős Problem #1139

A claim by Przemek Chojecki, gavinsherry, Liam Price and Terence Tao with GPT-5.2 Pro and GPT-5.5 Pro (19 Jun 2026) that has not yet been verified: candidate full solution.

Problem
Erdős problem #1139 (number theory, primes); current status: open
Credit
Przemek Chojecki, gavinsherry, Liam Price and Terence Tao with GPT-5.2 Pro and GPT-5.5 Pro · AI working with humans
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-06-19 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
UN-
VERIFIEDPer Tao's AI-contributions wiki

The problem

Let 1≤u1<u2<⋯1\leq u_1<u_2<\cdots be the sequence of integers with at most 22 prime factors. Is it true that

lim sup⁡uk+1−uklog⁡k=∞?\limsup \frac{u_{k+1}-u_k}{\log k}=\infty?

Statement quoted from erdosproblems.com/1139, maintained by Thomas Bloom. Original source: [Va99,1.4]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Humans Outcome
26 Jan-19 Jun, 2026 AI working with humans GPT-5.2 Pro, GPT-5.5 Pro Przemek Chojecki, gavinsherry, Liam Price, Terence Tao ⚪ Candidate full solution

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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