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 
.
