In OpenAI's preprint Integer multiplication below n log n (PDF): https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf

Yes, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More on YouTube (29:10, chapter 9): https://www.youtube.com/watch?v=6Tz3kgHKZ_s&t=167s

Integer multiplication below n log n | OpenAI Math, Explained

Integer multiplication below n log n

One of the 372 results in OpenAI's public mathematics catalogue, filed under theoretical computer science: 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.

Also known as integer multiplication beyond n log n; faster than n log n multiplication

In one line: OpenAI reports a result titled ‘Integer multiplication below n log n’ (theoretical computer science) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More, is on YouTube.

What OpenAI claims

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^1-κ), with κ=2^-182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen n log n optimality conjecture in the ordinary multitape bit model.
OpenAI, in its public catalogue of maths results

Paper

Integer multiplication below n log n
23 September 2026 · Preprint, not peer reviewed · PDF
OpenAI's abstract We give a deterministic algorithm that multiplies two n-bit integers in O(n(lg n)^1-κ) worst-case time, with κ=2^-182, on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the n log n optimality conjecture of Schönhage and Strassen in this model.

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 OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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?

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^1-κ), with κ=2^-182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen n log n optimality conjecture in the ordinary multitape bit model.

Where is the proof?

In the paper, Integer multiplication below n log n (PDF).

Is there a video?

Yes, on YouTube (29:10, chapter 9).

Integer multiplication below n log n

What OpenAI claims

Paper