OpenAI research collection · 1 manuscript · Source checked 2026-10-19
The idea in plain language
The paper claims a method for multiplying two very long integers that beats a longstanding proposed speed limit in a precise model of computation. The theoretical improvement is extremely small.
Why it matters
This would revise our understanding of computational limits. It is not evidence that ordinary arithmetic software immediately becomes faster.
Read the technical source claim
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^{1-\kappa}), with \kappa=2^{-182}, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen nlog n optimality conjecture in the ordinary multitape bit model.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
Integer multiplication below n log n
We give a deterministic algorithm that multiplies two n-bit integers in O(n(\lg n)^{1-\kappa}) worst-case time, with \kappa=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 nlog n optimality conjecture of…
Full source abstract
We give a deterministic algorithm that multiplies two n-bit integers in O(n(\lg n)^{1-\kappa}) worst-case time, with \kappa=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 nlog n optimality conjecture of Schönhage and Strassen in this model.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.