Proved in 1933. If 
is an odd prime or 
and 
is any positive integer,
then there is a Hadamard matrix of order
 |
where 
is any positive integer such that 
. If 
is of this form, the matrix can be constructed with a Paley
construction. If 
is divisible by 4 but not of the form (1), the Paley
class is undefined. However, Hadamard matrices
have been shown to exist for all 
for 
.
