// Workers AI · dad joke modeWhat did Williamson Conjecture say? "I'm just guessing".

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 called Williamson matrices if their entries are $\pm 1$ 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 by Dragomir Ž. Ðoković through an exhaustive computer search, which demonstrated that Williamson matrices do not exist of order $n=35$. In 2008, the counterexamples 47, 53, and 59 were additionally discovered.

Following the negative result of Ðoković, which ruled out the existence of Williamson matrices of order $n=35$, it was shown in 2019 that relaxing the symmetry and circulant requirements nevertheless permits a Hadamard matrix of this block form to exist for that order. One such instance is given by the sequences

a = --+--+--+++-+----+-+++--+--+-------
b = +---+---+-++-+--+-++-+---+---++++++
c = -++---+-+--+--++--+++++-+-+++++-+--
d = +----+++-+-+--++--+-+-+++----++---+

with the associated matrices defined by

A = circulant(a)
B = circulant(b)
C = fliplr(circulant(c))
D = circulant(d)