Combinatorics·OpenAI Math·Result 164
Hindman’s finite sums and products conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under combinatorics: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of Hindman’s finite sums and products conjecture (combinatorics) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 19 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 6:13 · watch from there on YouTube →
What OpenAI claims
Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.
Paper
Monochromatic finite sums and products in the positive integers
OpenAI's abstract We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k, there is a k-element set whose nonempty subset sums and nonempty subset products all have the same color.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.
- Where is the proof?
- In the paper, Monochromatic finite sums and products in the positive integers (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 19).
Also in
- Chapter 4 of 10 in Borsuk's Conjecture Fails in Dimension 9, Sidorenko's Conjecture and 8 More · starts at 1:02 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 19 from 6:13 · description, sources and chapters →
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