About the problem
A Hadamard matrix is a square matrix all of whose entries are \(\pm 1\) and whose rows are mutually orthogonal. Beyond trivial cases, the order of such a matrix must be a multiple of four. The Hadamard conjecture states that a Hadamard matrix exists for every such order. It remains open.
Mathematicians have chipped away at finding examples of Hadamard matrices of larger and larger order. The smallest case for which no matrix is known is \(668\). The previous smallest unknown case was \(428\), resolved in 2004 by Kharaghani and Tayfeh-Rezaie. New cases typically require somewhat clever and novel constructions.
This problem asks for a Hadamard matrix of order \(668\). As a warm-up, we ask for a matrix of order \(428\).
Attempts by AI
We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.
AI prompts
Warm-up
Find a Hadamard matrix of order 428. Provide your solution as a csv.
Full problem
Find a Hadamard matrix of order 668. Provide your solution as a csv.
Mathematician survey
The author assessed the problem as follows.
a majority of those working in a subfield (≈100)
5–10
1–4 weeks
moderately interesting
in a standard specialty journal
somewhat likely: an especially innovative solution could, but is hardly guaranteed
95–99%