A Hadamard matrix of order 668

The problem

A Hadamard matrix of order $n$ is an $n \times n$ matrix of $\pm 1$ entries whose rows are mutually orthogonal, so that $HH^{\mathsf T} = nI$. They can exist only when $n$ is 1, 2 or a multiple of 4, and the Hadamard conjecture says they exist for every multiple of 4. Since order 428 was found in 2004 by Kharaghani and Tayfeh-Rezaie, the smallest order for which no Hadamard matrix was known has been 668.

What AI did

On 12 August 2026 Levent Alpöge, a researcher at Anthropic, posted an encoded string that, according to Epoch AI, contains Hadamard matrices for every order up to 2000 for which none was previously known, 668 among them. The work is credited to three people and Claude. Alpöge added the same day that "there's a search (and an actual idea, which i don't claim!!) involved". No paper describing the method has been published.

How it was checked

On 13 August Ion Nechita reported checking that the entries are $\pm 1$ and that $HH^{\mathsf T} = nI$. A Hadamard matrix is fully checked by that computation, so the existence claim for order 668 is on firm ground.

Limitations

How the work was divided between the people and Claude is not clear, and Epoch AI calls its own label provisional (its changelog says "solved by AI", its problem card says "human + AI"). The Hadamard conjecture itself remains open. Problem description paraphrased from Epoch AI (CC BY 4.0).

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