Williamson conjecture
# Williamson conjecture

In combinatorial mathematics, specifically in combinatorial design theory and combinatorial matrix theory the Williamson conjecture is that Williamson matrices of order $n$ exist for all positive integers $n$. Four symmetric and circulant matrices $A$, $B$, $C$, $D$ are known as Williamson matrices if their entries are $\pm 1$ and they satisfy the relationship

    $A^{2}+B^{2}+C^{2}+D^{2}=4nI$

where $I$ is the identity matrix of order $n$. John Williamson showed that if $A$, $B$, $C$, $D$ are Williamson matrices then

    ${\begin{bmatrix}A&B&C&D\\-B&A&-D&C\\-C&D&A&-B\\-D&-C&B&A\end{bmatrix}}$

is an Hadamard matrix of order $4n$. It was once considered likely that Williamson matrices exist for all orders $n$ and that the structure of Williamson matrices could provide a route to proving the Hadamard conjecture that Hadamard matrices exist for all orders $4n$. However, in 1993 the Williamson conjecture was shown to be false via an exhaustive computer search by Dragomir Ž. Ðoković, who showed that Williamson matrices do not exist in order $n=35$. In 2008, the counterexamples 47, 53, and 59 were additionally discovered.

