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

$\left[\begin{array}{cccc}A& B& C& D\\ -B& A& -D& C\\ -C& D& A& -B\\ -D& -C& B& A\end{array}\right]$

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.