Williamson conjecture
In combinatorial mathematics, specifically in combinatorial design theory and combinatorial matrix theory the Williamson conjecture is that Williamson matrices of order exist for all positive integers.
Four symmetric and circulant matrices,,, are known as Williamson matrices if their entries are and they satisfy the relationship
where is the identity matrix of order. John Williamson showed that if,,, are Williamson matrices then
is an Hadamard matrix of order.
It was once considered likely that Williamson matrices exist for all orders
and that the structure of Williamson matrices could provide a route to proving the Hadamard conjecture that Hadamard matrices exist for all orders.
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. In 2008, the counterexamples 47, 53, and 59 were additionally discovered.
