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 $$\pm1$$ and they satisfy the relationship


 * $$A^2 + B^2 + C^2 + D^2 = 4n I$$

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.