Integer multiplication below n log n
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Is there a uniform exact deterministic multitape Turing-machine algorithm multiplying n-bit binary integers in O(n(log $n)^{1-2^{-182}})$ bit operations, for all sufficiently large n?Short English statement
Is there a uniform exact deterministic multitape Turing-machine algorithm multiplying n-bit binary integers in O(n(log $n)^{1-2^{-182}})$ bit operations, for all sufficiently large n?Sources
- S1 Original question source or mathematical context. https://mccurley.org/papers/open.ps.gz.
- S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2024/40.
- S3 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for CS-138
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 109; 2026-09-23).
Integer multiplication below n log n
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 109; 2026-09-23), the asserted result is that exact multiplication on a fixed multitape finite-alphabet Turing machine takes $O(n(\log n)^{1-2^{-182}})$ worst-case time.Mathematical statement of the claimed result
Manuscript 1.
For every $n\ge1$ and two binary words of length $n$, the claim supplies one fixed deterministic Turing machine, with fixed finite alphabet and fixed number of one-dimensional tapes, that outputs the exact product of their nonnegative integer values in worst-case time \[ O(n(\lg n)^{1-\kappa}),\qquad\kappa=2^{-182}. \] The convention is $\lg n=\max\{\lceil\log_2n\rceil,1\}$. In the same time scale, fixed machines produce quotient and remainder for a nonzero divisor and the integer square root, with padded binary outputs of length $n$ allowed. These are bit-operation bounds, not unit-cost real arithmetic.Outline of the argument
For manuscript 1 (family 109; 2026-09-23), two finite linear networks exchange data banks while restoring all auxiliary values. Subspace labels make representation changes cheap on array wires, and the rank saving survives recursive transforms; exact reconstruction and tape-stream accounting give the bit-time improvement.Formal verification
No Lean formalization is listed at this snapshot for manuscript 1 (family 109; 2026-09-23).This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Integer multiplication below nlog n, 2026-09-23; citation key
OAI:Integer-multiplication-below-n-log-n-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.